# 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 = bx, then logb y = x

Read logb 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 bx is the logarithm in the equation logb 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 of the expression log5 25 ?

2 is the logarithm of the expression log5 25. Why? In the expression, we can see that the base is 5.

Therefore, ask yourself, "5 to the power of what is equal to 25?"

Since 52 = 25, log5 25 = 2.

What is the logarithm of the expression log2 16 ?

Since 2 to the fourth power is equal to 16, log2 16 = 4.

## What is a common logarithm?

A common logarithm is a logarithm that uses base 10. Therefore, the expression logb 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

What is log10 1000 ?

Since 10 to the third power  = 1000,  log10 1000 = 3.

## Writing in logarithm form

Write 49 = 72 in logarithm form

Write the definition

If y = bx, then logb 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 = bx, then logb y = x

Substitute

If 1/8 = (1/2)3, then log1/2  1/8 = 3

The logarithmic form of 1/8 = (1/2)3 is log1/2  1/8 = 3

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