Logarithms
So the two sets of statements, one involving powers and one involving logarithms are equivalent.
In the general case, we have:
Key Point
if x = a^{n} then equivalently log_{a} x = n
log_{a} a = 1
We can see from the Examples above that indices and logarithms are very closely related. In the same way that we have rules or laws of indices, we have laws of logarithms. These are developed in the following sections.
There are two main types of logarithms that we commonly use: common logarithms and natural logarithms.
1. Common Logarithm:
2. Natural Logarithm:
When you multiply two logarithmic values, it's like adding their individual logarithms:
log_{b}(mn) = log_{b}m + log_{b}n
Dividing two logarithmic values is like subtracting their individual logarithms:
log_{b}(m/n) = log_{b}m  log_{b}n
The logarithm of a product with a rational exponent is equal to the exponent times its logarithm:
log_{b}(mn) = n log_{b}m
Switching between different bases is done by dividing the logarithm in the original base by the logarithm in the new base:
log_{b}m = log_{a}m / log_{a}b
Inverting the logarithm's base is done by taking the reciprocal of the logarithm in the new base:
log_{b}a = 1 / log_{a}b
If f(x) = log_{b}(x), then the derivative of f(x) is given by:
f'(x) = 1 / (x ln(b))
The integral of a logarithmic function is given by:
∫log_{b}(x)dx = x(log_{b}(x)  1/ln(b)) + C
 log_{b}b = 1
 log_{b}1 = 0
 log_{b}0 = undefined
We can use logarithms to solve equations where the unknown is in the power. Suppose we wish to solve the equation 3^{x} = 5. We can solve this by taking logarithms of both sides. Whilst logarithms to any base can be used, it is common practice to use base 10, as these are readily available on your calculator. So, log 3^{x} = log 5
Now using the laws of logarithms, the lefthand side can be rewritten to give
x log 3 = log 5
This is more straightforward. The unknown is no longer in the power. Straightaway
If we want, this value can be found in a calculator.
Example 1: Suppose we wish to find log_{2} 512.
This is the same as being asked ‘What is 512 expressed as a power of 2 ?’
Now 512 is 2^{9} and so log_{2} 512 = 9.
Example 2: Suppose we wish to find log_{8} 1/64.
This is the same as being asked ‘What is 1/64 expressed as a power of 8 ?’
Now 1/64 can be written 64^{−1}. Noting also that 8^{2} = 64 it follows that
using the rules of indices. So log_{8} = 1/64 = 2
Example 3: Suppose we wish to find log_{5} 25.
This is the same as being asked ‘What is 25 expressed as a power of 5 ?’
Now 5^{2} = 25 and so log_{5} 25 = 2.
Example 4: Suppose we wish to find log_{25} 5.
This is the same as being asked ‘What is 5 expressed as a power of 25 ?’
We know that 5 is a square root of 25, that is 5 = √25. So 25 ^{1/2} = 5 and so log_{25} 5 = 1/2.
Example 5: Consider log_{2} 8. We are asking ‘What is 8 expressed as a power of 2 ?’ We know that 8 = 2^{3 }and so log_{2} 8 = 3.
What about log_{8} 2 ? Now we are asking ‘What is 2 expressed as a power of 8 ?’ Now 2^{3} = 8 and so 2 = ∛8 or 8^{1/3}. So log_{8} 2 =1/3.
We see again
1. Why do we study logarithms? 
2. What is a Logarithm? 
3. What are the types of logarithms? 
4. What are the rules for logarithmic operations? 
5. How can logarithms be used to solve equations? 

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