Logarithm is the mathematical tool or function. By the use of logarithm, we find the power of the base to find out the required result. Like, if we need to find out the power of 4 to get the result 64, its clear that we need 4x4x4 i.e. 43. Here 4 is the base, 3 is the power and 64 is the result.

Simply, we can say that by the use of logarithm, we get the answer of the question that " What exponent do we need for one number to become another number"?

Example: How many 3s do we multiply to get 81.Solution: We need
3 x 3 x 3 x3 = 81, so if we multiply 3 four times with itself, we get 81. Hence logarithm is 4.

## Logarithm Definition

Exponents: Exponents and logarithm are opposite of each other, just like
multiplication and division. Exponents undo logarithm. Technically we say, exponents are inverse of logarithm.
Example: xa = y where a is the exponent of base x.

Exponents indicates how many times we use the number in multiplication. This is a quantity indicates the power.
Example:
24 = 16, so we multiply 2 four times with itself to get 16 (i.e. 2 x 2 x 2 x 2)
Here 4 is the exponent and 2 is the base.

Exponent is a variable. Exponents are positive and negative numbers, complex number, rational numbers etc. In an expression they are denoted by superscripts.

Logarithm: If we have the expression ax = y, here a is the base, y is the result
and x is the exponent.
Now we can say that " logarithm of a number y with the base a is the power (exponent) to which a has to be moved to get y". Generally we denoted logarithm by log.

Now if we have 24 = 16
or 2 x 2 x 2 x 2 = 16
Now we can write log216 = 4, so we say log base 2 of 16 is 4.

So to conclude, in logarithm, basically we deal with three things/ terms:
• Base: The number we are multiplying.
• Exponent: How many times we use the number to multiplication.
• Result: The number we want to get i.e. the result we want to get.
Example: What is log6(216) = ?
Solution:
We are trying to know about " how many 6s need to be multiplied
together to get 216".
So, 6 x 6 x 6 = 216, we need three 6s.
Then log6(216) = 3
In other words the log of y to the base a is the solution x of the equation

## Logarithm Formula

In general case if we have ax = y, here a is the base, y be the result and x is the exponent, then we can also express it in terms of logarithm and denoted by,
loga y = x
and defined as "log base a of y is equal to x" or "logarithm of y with base a is x".

In general, we have "log" with no base,it means, log with the base 10 i.e. log10 like,
log10 10000 = 4 or rewrite log 10000 = 4,

this usually means that the base is really 10.
On calculator its the "log" button.
But sometimes the base of the logarithm is not "10". So we use some formula to help in working with different bases, it may be numbers or variables. But bases of the logarithm(log) are never negative. So we use the below formula for different bases:
loga y = x.

## Logarithm Properties

Logarithm holds following identities for any positive a $\neq$ 1 and positive number x and y:
Property 1: loga (xy) = loga x + loga y
Multiplication inside the log can be turned the addition out side the log.
Proof:
Let loga x = A and loga y = B
then x = aA and y = aB
If we operate
xy = aA . aB = a(A+B)

Taking loga both sides, then
loga (xy) = loga a(A+B)
or
loga (xy) = (A+B) logaa
or
loga (xy) = A + B (since logaa = 1)
Replace the values of A and B, then

loga (xy) = loga (x) + loga (y)

Property 2: loga $\frac{x}{y}$ = loga x - loga y
Division inside the log can be turned into subtraction outside the log.

Proof:
Let loga x = A and loga y = B then x = aA and y = aB
Now $\frac{x}{y}$ = $\frac{aA}{aB}$
= a(A-B)
Takine log both sides,
loga $\frac{x}{y}$ = loga a(A-B)
= (A - B) logaa
= (A - B) (Since logaa = 1)
Replace values of A and B, then

loga $\frac{x}{y}$ = loga x - loga y

Property 3:
Change the base property i.e. if loga (x) = $\frac{logb x}{logb a}$
Proof:
Let y = loga (x)
or x = ay
Taking log both sides to base b, then
logb x = y logb a
or

## Logarithm Examples

Example 1: What is log10(100000) = ?
Solution:
We know that 105 = 100000
So an exponent of 5 is needed to make 10 into 100000. Hence
What is log10(100000) = 5.

Example 2: What is log8(512) = ?
Solution:
It is clear that 83 = 512
So if we take exponent of 3 base 8, we get 512, hence
log8(512) = 3.

Example 3: Convert "46 = 4096" to the equivalent logarithmic expression.
Solution:
To convert "46 = 4096", the base 4 remains same but 6 and 4096 switch their
sides. Then we get,
log4(4096) = 6.

Example 4:
Convert "log7(343) = 3" to the equivalent exponent expression.
Solution:
To convert "log7(343) = 3" the base 7 remains same but 343 and 3 switch their
sides. So we get,
73 = 343.

## Logarithm Problem

Problem 1 : Evaluate log2 16 - log28 + log2 2 = ?

Problem 2: What is log4(256) =?

Problem 3 : Which of the following statements is not correct?
(A) log33 =1
(B) log ( 2 + 5) = log ( 2 x 5 )
(C) log21 = 0
(D) log ( 2 x 3 ) = log 2 + log 3