# Explainer: Laws of Logarithms

In this explainer, we will learn how to use the laws of logarithms to simplify logarithmic expressions.

Recall that given a positive number (provided it is ) and any other positive number , the logarithm of to base is the number such that

We write this as . The laws of logarithms tell us how logarithms convert products and quotients to sums and differences and also how a quotient of logarithms can be interpreted as a logarithm.

### Laws of Logarithms

For a fixed base , where , and any positive numbers , and ,

1. products: ,
2. quotients: ,
3. powers: for any real exponent ,
4. change of base: .

We will remark on (4) briefly. Notice that it can also be read as by setting . This “change of base” law comes from the law of exponents that states that since raising to the expression on the left-hand side of the identity above gives us

This says that the number satisfies , which is exactly what is required for this number to be .

### Example 1: Using the Laws of Logarithms to Evaluate Expressions

Calculate .

Applying the logarithm law for quotients, we have

Note the strategy: Apply the logarithm laws before simplifying the numeric expressions and looking for further simplifications.

### Example 2: Using the Laws of Logarithms to Evaluate Expressions

Find the value of without using a calculator.

Applying the logarithm law for sums and then differences in turn, we have

### Example 3: Using the Laws of Logarithms to Evaluate Expressions

Find the value of without using a calculator.

We will reduce this to a quotient of logarithms by first applying the laws for sums and differences to simplify the numerator and denominator:

Finally, we will apply the law that says that this itself is a logarithm:

Often, the trick is to recognize the laws “in reverse”.

### Example 4: Using the Laws of Logarithms to Find Equivalent Expressions

Which of the following is equal to ?

Of course, refers to . We start by simplifying the numerator, using the power law of logarithms:

Applying the sum law of logarithms to the denominator gives

So,

The correct option is (C).

Remember that the change of base law has two different statements. Note the form we use here.

Simplify .