How To Evaluate Logarithms - Basic Mathematics

How to evaluate logarithms

This lesson will show how to evaluate logarithms with some good examples. Study the first example below carefully.

Examples showing how to evaluate logarithms

Example #1:Evaluate log8 16

Write an equation in logarithmic form

log8 16 = x

Convert the equation to exponential form16 = 8x

Write each side using base 2.24 = (23)x24 = 23xSince the base or 2 is the same,   24 = 23x   if 4 is equal to 3x

If 4 = 3x, then  x = 4/3Therefore, log8 16 = x = 4/3

Example #2:Evaluate log10 1000

Write an equation in logarithmic form

log10 1000 = x

Convert the equation to exponential form1000 = 10x

Write each side using base 10.103 = (10)xSince the base or 10 is the same,   103 = 10x   if 3 is equal to xTherefore, log10 1000 = x = 3

Example #3:Evaluate log64  1/16

Write an equation in logarithmic form

log64 1/16 = x

Convert the equation to exponential form1/16 = 64x

Write each side using base 4.1/42 = (43)x4-2 = 43xSince the base or 4 is the same,   4-2 = 43x   if -2 is equal to 3x

If -2 = 3x, then  x = -2/3Therefore, log64  1/16 = x = -2/3

Example #4:Evaluate log5 (-25)

Write an equation in logarithmic form

log5 (-25) = x

Convert the equation to exponential form-25 = 5x

No matter what x is, you could never get 5x to equal to -25!

The logarithm of a negative number does not exist as a real number!

How to evaluate logarithms when it is not possible to write each side of the equation with the same base.

Example #5:Evaluate log3  10

Write an equation in logarithmic form

log3  10 = x

Convert the equation to exponential form3x = 10

Now, as you can see it is not possible to write each side with a base of 3 since it is very hard to rewrite 10 with a base of 3.

What we can do is to take the common logarithm of each side

log10 3x = log10 10

What is log10 10 equal to?

In logarithmic form, log10 10 is log10 10 = y

In exponential form, log10 10 = y is 10y = 10 or 10y = 101

So y  = 1 and log10 10 = 1

Substitute 1 for log10 10 in log10 3x = log10 10

We get log10 3x = 1

x log10 3 = 1

x = 1/(log10 3)

x = 1/0.477

x = 2.096

log3  10 = 2.096

Now study carefully the following figure with summarizes the concept

Exponential and logarithmic functions

How to simplify logarithms

Properties of logarithms

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