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Chapter 12: Problem 95

Describe the product rule for logarithms and give an example.

Short Answer

Expert verified
The product rule for logarithms states that the log of the product of two numbers is equal to the sum of the logs of its factors, expressed in the form: \( log_b(M * N) = log_b(M) + log_b(N) \). An example would be \( log_2(4 * 2) = log_2(4) + log_2(2) \), where both sides evaluate to 3, validating the rule.

Step by step solution

01

Product Rule Explanation

The product rule is one of the basic properties of logarithms. It states that the logarithm of the product of two numbers is equal to the sum of the logarithms of those two numbers. If we have two numbers, M and N, and we take their product, we attribute a base b to it, the rule could be written as: \( log_b(M * N) = log_b(M) + log_b(N) \). This rule holds true for all base logarithms.
02

Providing an Example

Now let's illustrate this with an example using base 2 logarithm. Let's consider 4 and 2 for M and N respectively. Then the statement of the product rule, \( log_b(M * N) = log_b(M) + log_b(N) \), becomes, \( log_2(4 * 2) = log_2(4) + log_2(2) \). Computing the values on both sides of the equation we get, \( log_2(8) = 2 + 1 \). As log of 8 in base 2 equals 3, and the sum on the right side is also 3. Since both sides are equal, it validates our product rule.

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