Simple Logarithm Tip

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Maths expression may at times look challenging, but a bit of a thought may make it otherwise.

Logarithm is always an exciting topics to new learners.

With the "log" coming into the maths expression, one will be confused.
Definitely!

But do rest assure, as the tip below shows.

Maths Tip

e^{ln y} = y

Proving this:
"Natural log" both sides will give  ln e^{ln y} =  ln y

Applying the law that ln aⁿ = n ln a,  and that ln e = 1, you will notice that the above mathematical expressions are true and equal.

NOTE:
This tip applies to "log" too.

10^{log y} = y


Hope this helps.

:-)

Logarithm Operation Explained (2)

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Logarithm operator exists as a mathematical tool to allow us to convert a number to another form in terms of base and power.

Can you make 0.5 in terms of 10^y or 5^y ?

The above is a maths question that can use logarithm to solve.

The application requires the Power rule that states that log_kD^m is equivalent to mlog_kD.
With that knowledge, to convert the above question of 0.5 to various base, we simply log the 0.5 to its respecive base.

Let's go for the base 10.

Original:   0.5 = 10^y

Performing "log" on both sides:
log₁₀ 0.5 = log₁₀10^y 
-0.301      = y  log₁₀10
-0.301      = y

or another way to put the outcome is 0.5 = 10^-0.301

We have managed to convert the original number of 0.5 to one with base 10 and a power (index) of "-0.301".

We can also similarly do the same for a base of 5. =>  0.5 = 5^y
Here we just "log" to base 5.

log₅ (0.5) = log₅ 5^y = y
-0.431       = y

Thus 0.5 = 5^-0.431

From the 2 examples above, you can see the wonders of having logarithm as a conversion tool.
Once you understand this principles, you will appreciate logarithm.

Cheers!
:D

Logarithm Operation Explained

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Everyone knows what happen when we add a number to another.

We also know what happen when we do subtraction.

There is also no problem with multplication or division of numbers.

These are all simple mathematical operations that are basic.

What about doing a logarithmic operation in math?
This is a slightly complex but interesting question.

When we do a logarithmic operation on a number, what we actually do want out of the mathematical process is the power or index with reference to a base number.

10⁵ has a base number of 10 and index of 5.


Doing a "log" of the above will reveal an answer of 5.
This is the power after doing a "log" operation.

Thus, when anyone does a logarithmic operation on a number (or expression), he is trying to find the index with respect to a base reference.

log 10⁵ = 5.

Note: 
Full written log expression is "log_kP".   When the "k" is left out, it implies that k = 10.
And log 10 = 1.

I hope the above can explain why we do logarithmic operation and its significant.

There must be a reason for each math operation, otherwise we will be learning and doing some insane process on earth!

Maths Is Interesting!

:-D

Math Challenge 18

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From the logarithm equation below, can you determine what is the expression for y?

 log y = x + log x

You may give your answer in the comment section and also help explain how you get it (for the sake of sharing).

Thanks ^^

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Power Law of Logarithm Explained

Logarithm study has a few formulae that are important and key to solving math questions.

By remembering them , you will be in line to solve logarithmic problems and, maybe, fast too.

However, what if you forget them?
Does it mean that you are not able to solve the question regardless of speed?

Do not despair.
As long as you are able to manage the 2 basic laws in logarithm, you are safe.

The product law and the quotient law are must for any students.

Why do I say that?

Let us take the Power Law and do a review.

n log x = log X ⁿ

Why is it so? What if you forget this law? Any problem?

These are the very queries any new learners exposed to logarithm will ask.

First allow me to go through the product law of logarithm.
log (XY) = log X + log Y

Here you see that the product of "X" and "Y" in logarithmic operation, becomes a "sum"of the individual logarithms.

4 log Y = log Y + log Y + log Y + log Y (adding up 4 of the log Y)

Using product law, you know that these 4 terms can be combined to log (Y x Y x Y x Y).

log (Y x Y x Y x Y) = log Y⁴

Now, you see, through the product law, you are able to equate the 4 log Y into log Y⁴ ,
meaning, 4 log Y = log Y⁴.

You see that you did not utilise the Power Law here,and yet is able to form this formula!
Amazing isn't it.

What is the message here?
The message is that, when you have the basic understanding in logarithmic principles, you will be able to twist and turn any given problems to come out a solution.

You had used the basic product law to discover this unique Power Law.
It is similar to other laws and also can be expanded to cover other maths topics too.

Do enjoy maths.
Do discover more exciting twists it presents wwith a bit of thinking.

Happy learning :-)

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Logarithm | Common Mistake

A common mistake occurs normally during simplification to a single logarithm term.

Question like,

" Simplify log X - log Y + log Z into a single term "

catches many students who are careless.

What is the error or mistake made?
- Doing the solving at one go when not familiar with the logarithmic rules
- Sign interpretation

Wrong answer given: log X/(YZ)

Correct answer: log XZ/Y

"log Z" is commonly taken to follow the previous log term, which is, "- log Y ".
Since "- log Y " causes the "Y" to be a denominator, "Z" is also taken to be a denominator too!

This is a mis-cue. A mental slip, mathematically.

Advise:
Look at the sign carefully before jumping to conclusion.
Go slow in the combination to a single log term.

Remember the idiom: "Slow and steady wins the race"

You can apply this to log simplification when you are new to it.

Cheers! :-D

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Key Differences Between Cos and Log Operations

It has been mentioned in previous posts that "cos" and "log" are actually mathematical operators.

They cannot exist alone. They need partners.

For "cos", it needs angle.
For "log", it requires a number.

They operate on something.

So what is their difference since they are both operators?

"cos" changes angle information to number (ratio).

"log" operates on number to get another number.

Their applications, thus, differ in the above matter.

NOTE: "cos" here refers to any trigonometrical functions.

"cos" is useful in getting length or angle information.

"log" can be used to find the value of the power needed.
Example: To find 2^x = 5

Knowing the various mathematical tools, we can apply the correct ones to solve specific problems.

Without understanding the underlying usage and concepts, we will be at a lost to which tools to use.

Math is, therefore, wider in nature than just calculation.
Keep learning....

:-)

Division of x^n with y^n

Confusion does happen when you are bombarded with many numbers, exponential and its likes.

After dealing with indices, logarithm and their multiplication and division, the brain will sort of tangle up and produces weird happenings.

Take 2 examples below:

1) x^n / y^n ==> x /y

2) log x^n / log y^n ==> (log x) / (log y)

By looking at the first example, you may find nothing wrong.
Since the power "n" is similar for the numerator and denominator, you can do the normal cancellation as you do for "mx / my" = x/y.

However, something may tell you that something is amiss.

While "mx / my" is truely x / y, this is because mx means m times x.
There are "m" number of x that are ADDED up.

For x^n, it means "x" is multiplied by itself n times. (or x times x times x times x ....)

Thus x^n is not equal to xn.
The truth of "cancellation" is that since a / a = 1, and this "1" is not required to be written, the disappearance seems to be "cancellation".

Let me explain further with an example(A).
ax / ay = (a/a)(x/y) = (1) (x / y) = x / y. The "1" disappeared and seems to be cancelled off.

The mistake made in Example 1 in the beginning, is the assumption that the powers "n" followed the concept of "ax" in example(A).

Correct answer for x^n / y^n = (x/y)^n. ==> The powers of n are not removed.

Now for the Example 2, at the beginning, it showed a similar cancellation of the powers "n".
But this time round, it can be said to be conditionally correct.

Why?

If the idea that similar "letter" of "n" in the power can be removed through cancellation, then the answer, although correct, is theoretically wrong.

However, if you know that using the Power Law of logarithm, log x^n can become "nlogx" and therefore, log y^n can also be "nlog y", the result of (log x) /(log y) can be rightfully considered correct, since the "n" is removed according to the idea that n/n = 1 and disappeared, or qualified for removal.

In summary, mistakes do happen when the concept of power (x^n) and pure multiplication (x times n) is not clearly understood.

Cancellation of "letters" or symbols in math expressions should be highlighted as a shortcut to removal due to being "1" that can be omitted in the written form.

This concept of "cancellation" is easy if you understand that it is because of a/a = 1.

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Common "log" mistake

Teacher: John, can you give the answer for X in this log X = log 6 question?

John: No problem. The value for X is simply 6.

Teacher: Correct! How did you get the answer?

John: It is easy. Just do it this way
(John wrote on the board). ==> X = (log 6) /log ==>X = 6.

Teacher: ?????

************************* What happened? ***********************

Logarithm or "log", in its abbreviated form, can be easily misunderstood.

What is this "log"?

Logarithm is an operation on a number that is the reverse of that for indexing a number.

Log_aX = Y ==> a^Y = X

From the above relationship, you will notice that "log" itself cannot stand alone.

That means "log" must come with a number or expression.

"Log" is an operator, like the "+" or "-".

What mistake did John made?

John mis-interpreted the "log" to be a variable!

It made him transfer the "log" over the equal symbol as though it is a number (or equivalent).
the "log" is thus, separated from the "X" that it should operate upon.

A common "log" mistake was made.

Correct answer:
The answer can be obtained through logically comparison, that is,
when log X = log 6, X is simply = 6.

The question may be simple, but if the learning is improper, the concept behind it may be drastically, wrong, even though the answer can be correct.

Learn well. Maths does not call for correct answer. It is the thinking behind it.

..... :-)

Log X Is Just A Number

I happened to come across an interesting mistake made by a maths student.

The mistake inter-mixed the principles of algebra with logarithm.

The maths question is to solve the value of x given the expression:

2+ log (5x - 1) = log 3x

The expression, after transferring the "log (5x -1)" to the right side of the "=", became

2 = log 3x - log (-5x + 1) !

Spotted the mistake done ?

Why was the "log (-5x + 1)" in that form?

The correct expression should be 2 = log 3x - log (5x - 1).

What actually went inside the student's mind was confusion between algebra and logarithm. He did not understand the concept of "logging" the (5x -1).

Log X is always a number!

Similarly log (5x - 1) is also a number.

Therefore log (5x - 1) moves as a number, same as in moving algebraic term.

If we have 2 + (x-a) = y, re-arranging the expression, gives us 2 = y - ( x-a).

The term "x-a" is taken as a whole, with change in the sign of (x -a) and not including that of the individual internal "a" and "x". This is basic algebra.

Moving log (5x - 1) is the same. Being a number, it operates equivalent to the algebraic manipulation.

The log (5x -1) is thus taken as a whole and sign change affects only the term as a whole. It does not affect the individual internal "5x" and "-1"!

Part of learning maths is following rules and principles.

The mistake made by the student was a reflection of correct algebraic change, but in the wrong sense. "Log" had converted the term into a number, and that was the mistake not captured.

Being careful with every steps taken in solving maths questions is a discipline one can treasure. This is one of the interesting part of doing maths!

<*_*>
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Ways To Improve Logical Learning Part Of Maths

In learning of maths, as with learning of other subjects, we need to understand our learning styles. The styles are namely, visual, auditory and kinesthetic.

Each of us has one particular dominant style. Knowing which one will thus serve us good. However, the learning style itself is still not enough, we need to also know the types of intelligence we possess.

The learning styles are used to gather information and ideas, through the senses, for the brain. But how do we process the information captured afterwards depends much on the intelligences that we also have.

In maths learning, what we need is the logical / mathematical intelligence.

This lets us do all sort of computations and comparisons that maths requires.

An example is the conversion of an expression in logarithmic form to its index form (and vice versa).

log_aY = X <==> a^X = Y

Though, this conversion seems simple enough, it is found that many students still have difficulties in converting the above. Do they lack this logical intelligence?

The answer is "maybe" and "maybe not".

They may have possess this logical intelligence, but has not yet enhanced it.

One way to offset this weakness in the logical comparison part of intelligence is to introduce the visual intelligence into maths learning.

What is this visual intelligence?

This is the "appearance" aspect of information processing. Here, colour, shapes, and the likes are linked up to the information.

Why introduce the visual intelligence?

The main reason is to bridge up the right and left brain, a theory that is now well-known and practiced throughout the learning and teaching communities.

Now, let's see how we can solve the logarithmic and index conversion through the use of the visual intelligence.



The examples above showed two very interesting ways to enhance the learning. :)

The first one is to replace and "beautify" the symbols with pictures or graphics to arouse retention ability.

The second one is to make use of colours to strengthen the symbols and their placements.

By seeing pictures and colours, although they are no way close to any maths topics, the learning of maths is greatly improved especially for the logical comparison portion.

Therefore, if possible, introduce as many "visuals" into maths as you can. They will make maths learning a totally new experience. Take care of intelligences and intelligences will take care of you.

Inspired ?

:=)

Indices & Logarithm - Less Mistakes with Proper Writing

Every symbols in mathematics serve certain purpose. The size of the symbols and numbers also represent unique meaning.

Many mistakes and confusions were made due to improper writing of the above two items that are very much key to mathematics.

Two of the topics that are commonly populated with writing errors are Indices and Logarithm.

Indices: a^x = Y

Logarithm: log_nA = K

From the above two examples, it is apparent that writing is crucial to give correct meaning to mathematics.

Writing therefore is an important skill while doing these two maths topics.

Discipline is needed here to maintain consistency.

What do I mean?

Let me give an example.

log₂ xy = log₂x + log₂y

The above is a correct expression written with proper size and positioning.

If writing discipline is not maintained, it may end up as:
log₂ xy = log₂x + log2y (size and position error)

Do you notice the change in meaning when size changes?

The subscript "2" representing "base 2" is conveyed as "2 times y"!

This is the result of a simple "slip" of the hand in writing the size of the "2".

Can this mistake be allowed?

I guess NO!

Why run the risk of marks being deducted or effort wasted in getting wrong answers?

Write properly is the only answer to less mistakes, if not at all.

If a^x = Y is written as ax = Y, how do we expect to get correct answers?

Here, you realise that maths learning is not just about mathematical concepts, human discipline and strength of concentration matters also.

Being able to write clearly and presentable is a noble capability that we should be proud to demonstrate to anyone. Strive to be focused and discipline when dealing with maths.

Happy writing! :P

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Graph In Logarithmic Scale

In graph plotting, we can have a choice to plot the raw data (from the math equation) directly using the conventional y- axis (vertical line) and x -axis (horizontal line) , or modifying the horizontal axis parameter to match the given math expression.

Which technique to use depends on our purpose.

If the objective is to present the data as it is, and revealing the trend the raw data ( y and x) gives regardless of its plotted profile, then the conventional y-axis and x-axis method suffices.

If the objective is to obtain a straight line graph from the non-linear (assumingly) math equation for analysis purpose, then the need to modify the horizontal axis parameter before plotting is required. Refer to this link for details on straight line conversion for non-linear math equations.

Example: y = log x

In direct plotting of the above expression, we will get a non-linear presentation as shown in Diagram 1 below. (The horizontal parameter is directly the "x" data.)



Diagram 1: Non-linear presentation of the y = log x (Horizontal axis: x)

When we desire to have a straight line graph, we can plot the horizontal axis with the modified data of "log x" to match the generic straight line equation of y = mx + c.

The below graph is plotted in Diagram 2 with the horizontal data as "log x". Here we should expect a straight line graph to appear.



Diagram 2: A straight line graph with modified horizontal axis of "log x" data.

** But note that this post will explain another method to achieve a straight line plot when the math equation involves the logarithmic operation. **

The disadvantage of the second method presented above involved the modification of the horizontal parameter which may results in confusion to the raw "x" data.

What this new plotting option involves is not the modification of the horizontal parameter but the re-adjustment of the division between the scales of the horizontal axis.

This re-adjustment of the horizontal scale results in the "Logarithmic Scale". This scale is a special scale that increases logarithmically while keeping the parameter as the raw "x" data in the math equation. The merit is the outcome is the desired straight line graph.

Let's see the y = log x expression plotted in the Logarithmic scale. (Diagram 3).



Diagram 3:
The straight line plot with Logarithmic scaling while keeping the raw "x" parameter.

Therefore, the math expression can be plotted using 2 techniques to achieve a straight line.

Logarithmic scale is commonly used in the plotting of frequency response of audio signal and filters. System stability is also analysed using logarithmic plots.

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Differences Between Logarithmic and Trigonometric Term

The mathematical expressions for logarithm and trigonometry looks similar.

Logarithm expression: Log A

Trigonometric expression: Cos A
(this has been randomly chosen; Sin A and Tan A is equally fine).

Both are numbers operated upon to give another meaning. However, they have their differences.

Logarithm

The number operated upon is pure number and is related to indices. That is, x^y = A

The main objective of "logarithming" is to find the power to a number (base).

Its application is more towards algebraic manipulation converting number to number in another form (power to base, and vice versa).

Another application is to determine the number of digits needed to get a desired number (this may be in digital electronics where the number of bits is needed to form a certain number of counts ==> 2ⁿ = 10 , logarithm is needed to find "n").

Trigonometry

The number operated upon is angle and is related to length.

The main objectives of trigonometric operations are to determine the amplitude of height of a frequency component and the lengths of the side of a triangle formed by the defined angle.

The result of doing a trigonometric function is a ratio of 2 lengths of a triangle. It translates angle information to information of the number ratio (of 2 sides).

Formula

Log (XY) = Log X + Log Y

Cos (A + B) = Cos A Cos B - Sin A Sin B

The formulae above are quoted to reflect the similarity but with slight differences.

Note that "Log" is with (XY) and "Cos" is with (A + B).

We should be careful not to inter-mix the concept and formulae as they serve different purposes.
This is the fun part in maths. The challenge of recognising the format and meaning is the other exciting part.

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Mental Logarithm Easily Computed

In our daily life, we encounter many maths computing which we do mentally. Some are simple addition and subtraction. Some involves more complex operation. But with breaking-down strategies, we are abale to simplify this mental process to simpler steps for the mind to handle.

Here I will discuss on the mental computation with maths operation like the indices and logarithm, that we may come also in some daily situations.

Below site an example. Read and enjoy it.

Example: 10^x = 8

To solve this example, we need to convert it into the logarithm form to move forward.

log 10^x = log 8 ---(A)

Applying the Laws of Logarithm, log Y^x ==> x log Y

Expression (A) above, become x (1) = x = log 8

However, we are stuck here! What is the answer for log 8? And mentally?

How to carry on with mental logarithm computation?

Mental logarithm computation can be easily processed if we can simply memorise a few number of the basic logarithm.

The basic numbers that we need to remember and suffice for common mental logarithm usage are:

• log 2 = 0.301


• log 3 = 0.477


• log 7 = 0.845

With the 3 basic logarithm numbers known, we can list out the first few log to see its usefulness:

• log 1 = 0


• log 2 = 0.301


• log 3 = 0.477


• log 4 = log (2 x 2) = log 2 + log 2 = 0.301 + 0.301 = 0.602


• log 5 = log (10 / 2) = log 10 - log 2 = 1 - 0.301 = 0.699


• log 6 = log (2 x 3) = log 2 + log 3 = 0.301 + 0.477 = 0.778


• log 7 = 0.845


• log 8 = log (2 x 2 x 2) = log 2 + log 2 + log 2 = 0.301 x 3 = 0.903


• log 9 = log (3 x 3) = log 3 + log 3 = 0.477 + 0.477 = 0.954


• log 10 = 1

With the above list, most of the logarithm numbers can be easily computed mentally.

But do note, however, that logarithm of prime number cannot be done using above method.
Examples are log 11, log 17, etc.

However, we should not be deterred from this as they are the minority and the list above serves to cover most of the logarithm solutions which can be easily done mentally.

:)

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How Are Indices and Logarithm Related

Is there any relationship between Indices and Logarithm?

To those unknown, Indices and logarithm are closely related to each other.
Their difference lies in the way they are presented or in their format.

Indices format: a^x = y

Logarithm format: log_aY = x

How do we select the appropriate format to use?
Which format to use depends on the initial maths problem or questions given, and what is the unknown to be solved.

If all else is given and the power of a number / expression has to be determined, select the "Logarithm" format. Power (x) = log_aY ==> directly obtaining the power x.

If the base "a" and the power "x" is known, then the Indices format can be used.

Example of Real - life Application:
In digital electronics engineering (where the base is 2 due to only two numbers present, 0 & 1), when the designer has to know the number of stages (x) to represent a count cycle of 8, he applies the logarithm format. log₂8 = x

For more information about the topics, click here to be directed to that post.

:-)

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Change of Base Method in Logarithm (Video)

There are times where the numerical base of the logarithmic term is not the one that we desire. We may need another value to match the other logarithmic terms.

In such instances, we need to change the base.

The video posted here shows the use of "Change of Base" method to solve logarithmic question involving different number base.



For more information and examples on this topics of changing numerical base, refer to this link.

:-)
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Logarithmic and Exponential Conversion (Video)

Logarithm and indices, or rather, the exponential form of logarithm, are closely linked,and served as a pair.

Maths learners of logarithm, thus, have to know and understand the translation of both forms (the log and exponential form).

The video here demonstrated to us how to convert the Logarithmic form to Exponential form.

This knowledge is a must for users of Logarithms (and Indices).

Solving Logarithmic Questions (Video)

Logarithm comes in two basic forms.
- Normal "log" to any number (integer) base,
- Natural "log" to the base of "e" (2.7132....)

The video posted here shows the steps in evaluating examples of logarithmic problems. Some knowledge on Natural Logarithm is needed, though.

Introductory Logarithm (Video)

Logarithm is very much part of our life.

An example is our hearing. Its response is logarithmic in nature.

This video shows the basics of Logarithm in maths studies.

It introduces the various elementary concept required in dealing with logarithmic questions.

Do note, however, that logarithm is closely related to another maths topics, Indices.

Thus, the video has some links to this topic on indices as well.

Cheers! Happy viewing.....



For more information and common mistakes in logarithm, click this link.

:-)