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 ,
- products: ,
- quotients: ,
- powers: for any real exponent ,
- 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 .
Answer
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.
Answer
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.
Answer
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 ?
Answer
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.
Example 5: Using the Laws of Logarithms to Simplify Expressions
Simplify .
Answer
The base change result is that
The product we have does not quite fit, but if we remember that , we can use the power rule:
So,