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.

To know about logarithm, first we know about exponents.
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.
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) = ?
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
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 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.
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)
loga (xy) = (A+B) logaa
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.

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) = $\frac{logb x}{logb a}$
Let y = loga (x)
or x = ay
Taking log both sides to base b, then
logb x = y logb a
y = $\frac{logbx}{logba}
loga (x) = $\frac{logb x}{logb a}$

Property 4: loga(x)n = n logax
An exponent on everything inside a log can be mover out front as a multiplier.
Let s = loga x
then as = x ( by the use of definition of logarithm)

Raising this equatio to the nth power, then we get
asn = xn
Rewriting above equation in logarithmic form with the base a,
loga xn = loga (asn)
= sn
= n loga x (since s = loga x )
In mathematics, we have to follow some logarithm rules:

(1) The log with a negative base does not exist i.e. log-a x does not exist.

(2) The log of a negative number does not exist i.e. loga (-x) does not exist.

(3) The log of zero does not exist i.e. loga 0 does not exist.

(4) The log of 1 is zero it means loga1 = 0.

(5) The log with base a and the number a is 1 i.e. logaa = 1.
→ Read More
The natural logarithm is the log with the base"e" i.e. loge , where e is the irrational constant approximately equal to 2.7182. Natural logarithm is defined for all real numbers as well as complex numbers. It is written as loge x or In x. On calculator it is "In" button.

The natural logarithm of a number x is the exponent to which e would have to be raised to equal x or simply it means how many times we need to use "e" in a multiplication to get the required result.
So if we have y = In a
$\Rightarrow$ ey = a
and In(e) = 1 since e0 = 1

Example: Evaluate In (e5).
By definition of natural logarithm we know that "In()" means the base e-log, so
In (e5) might be thought of as loge(e5). Thus
loge(e5) = y then y = 5.
So In (e5) = 5. → Read More
Inverse logarithm is also known as antilogarithm. The antilogarithm(or antilog) is the inverse of logarithm. If x is the logarithm of y then y is the antilogarithm of x i.e. the number of which a given number is the logarithm is said to be the antilogarithm and denoted by antiloga y = x $\Rightarrow$ ay = x.

Since the log(base 10) of 100 equals to 2, the antilogarithm of 2 is 100. To calculate the antilogarithm of a base 10 logarithm, take 10 to that power and to compute the antilogarithm of a natural logarithm, take 'e' to that power.

Thus finding the antilogarithm of a number is the same as finding the value for which the given number is the logarithm. If log10 y = 3, then 3 is the power to which one must raise the base 10 to obtain y, i.e. y = 103 = 1000. The determination of an antilogarithm is the reverse process of finding a logarithm.

If log2 8 = 3 then antilog23 = 23 = 8.

There is no significance of negative logarithm. But logarithm deal with multiplying. It is wll known that the opposite of multiplying is dividing.
Hence, "negative logarithm means how many times to divide by the number".

Example 1: What is log2 (0.5) = ?
Well we know that $\frac{1}{2}$= 2-1 = 0.5 so log2(0.5) = -1.

Example 2: What is log4 (0.0625) = ?
Since if we have 1x $\frac{1}{4}$ x $\frac{1}{4}$ = 4-2
So log4(0.0625) = -2.
If we have a complex number say z = a + ib, where i2 = -1 and indicate the imaginary quantity.
The logarithm of a complex number z is defined as every complex number u which satisfies the equation eu = z and log z = u.
The solution of the above relation is obtained by the formula eu = rei$\theta$ , where r = IzI and $\theta$ = arg z. Hence
u = log z = In IzI + i arg $\theta$.
IzI indicates natural logarithm of the real number IzI, and $0\leq \theta \leq 2\pi$ .
So log z = In r + i$\theta$ + 2n $\pi$ i ( n = 0, $\pm$1, $\pm$2, ......)
This is defined for all z$\neq$ 0.
If n = 0, then
log z = In r + i$\theta$.

Example 1: What is log10(100000) = ?
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) = ?
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.
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.
To convert "log7(343) = 3" the base 7 remains same but 343 and 3 switch their
sides. So we get,
73 = 343.
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