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 form

16 = 8^x

Write each side using base 2.

2⁴ = (2³)^x

2⁴ = 2^3x

Since the base or 2 is the same,   2⁴ = 2^3x   if 4 is equal to 3x

If 4 = 3x, then  x = 4/3

Therefore, log₈ 16 = x = 4/3

Example #2:

Evaluate log₁₀ 1000

Write an equation in logarithmic form

log₁₀ 1000 = x

Convert the equation to exponential form

1000 = 10^x

Write each side using base 10.

10³ = (10)^x

Since the base or 10 is the same,   10³ = 10^x   if 3 is equal to x

Therefore, 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 form

1/16 = 64^x

Write each side using base 4.

1/4² = (4³)^x

4^-2 = 4^3x

Since the base or 4 is the same,   4^-2 = 4^3x   if -2 is equal to 3x

If -2 = 3x, then  x = -2/3

Therefore, 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 = 5^x

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

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 form

3^x = 10

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

log₁₀ 3^x = log₁₀ 10

What is log₁₀ 10 equal to?

In logarithmic form, log₁₀ 10 is log₁₀ 10 = y

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

So y  = 1 and log₁₀ 10 = 1

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

We get log₁₀ 3^x = 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