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

**Example #1:**

Evaluate log_{8} 16

Write an equation in logarithmic form

log_{8} 16 = x

Convert the equation to exponential form

16 = 8^{x}

Write each side using base 2.

2^{4} = (2^{3})^{x}

2^{4} = 2^{3}^{x}

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

If 4 = 3x, then x = 4/3

Therefore, log_{8} 16 = x = 4/3

**Example #2:**

Evaluate log_{10} 1000

Write an equation in logarithmic form

log_{10} 1000 = x

Convert the equation to exponential form

1000 = 10^{x}

Write each side using base 10.

10^{3} = (10)^{x}

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

Therefore, log_{10} 1000 = x = 3

**Example #3:**

Evaluate log_{64} 1/16

Write an equation in logarithmic form

log_{64} 1/16 = x

Convert the equation to exponential form

1/16 = 64^{x}

Write each side using base 4.

1/4^{2} = (4^{3})^{x}

4^{-2} = 4^{3}^{x}

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

If -2 = 3x, then x = -2/3

Therefore, log_{64} 1/16 = x = -2/3

**Example #4:**

Evaluate log_{5} (-25)

Write an equation in logarithmic form

log_{5} (-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!

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

**Example #5:**

Evaluate log_{3} 10

Write an equation in logarithmic form

log_{3} 10 = x

Convert the equation to exponential form

3^{x} = 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

log_{10} 3^{x} = log_{10} 10

What is log_{10} 10 equal to?

In logarithmic form, log_{10} 10 is log_{10} 10 = y

In exponential form, log_{10} 10 = y is 10^{y} = 10 or 10^{y} = 10^{1}

So y = 1 and log_{10} 10 = 1

Substitute 1 for log_{10} 10 in log_{10} 3^{x} = log_{10} 10

We get log_{10} 3^{x} = 1

x log_{10} 3 = 1

x = 1/(log_{10} 3)

x = 1/0.477

x = 2.096

log_{3} 10 = 2.096