What is a logarithm?

What is a logarithm? In mathematics, the logarithm to the base b of a positive number y is defined as follows:

If y = b^x, then log_b y = x

Read log_b y as " log base b of y "

Just like we saw in the lesson about exponential function, b is not equal to 1 and b is bigger than zero.

The exponent x in the exponential expression b^x is the logarithm in the equation log_b y = x

What is a logarithm in easy terms?

Keep in mind that whenever you are looking for the logarithm, you are looking for an exponent, or the number that tells how many times the base is multiplied.

For example, what is the logarithm in log₅ 25 = 2?

2 is the logarithm of the expression log₅ 25. And it means that the base or 5 must be multiplied by itself 2 times to get 25.

Common logarithm

A common logarithm is a logarithm that uses base 10. Therefore, the expression log_b y = x becomes log₁₀ y = x

As a result, you are always looking for the number of times you multiply 10 to get y.

You can write the common logarithm as log₁₀ y or as log y

Writing in logarithm form

Write 49 = 7² in logarithm form

Write the definition

If y = b^x, then log_b y = x

Substitute

If 49 = 7², then log₇ 49 = 2

The logarithmic form of 49 = 7² is log₇ 49 = 2

Write 1/8 = (1/2)³ in logarithm form

Write the definition

If y = b^x, then log_b y = x

Substitute

If 1/8 = (1/2)³, then log_1/2  1/8 = 3

The logarithmic form of 1/8 = (1/2)³ is log_1/2  1/8 = 3