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Chapter 4: Problem 105

Describe the product rule for logarithms and give an example.

Short Answer

Expert verified
The Product Rule for logarithms states that for any positive numbers M, N, and a number b (b 鈮� 1), \(\log_b (MN) = \log_b M + \log_b N\). Example: \(\log_2 32 = \log_2 4 + \log_2 8 = 2 + 3 = 5\).

Step by step solution

01

Defining Logarithm Rules

The Product Rule for logarithms states that for any positive numbers \(M\), \(N\), and a number \(b\) (\(b \neq 1\)), if the base-b logarithm of \(M\) and \(N\) are known, then the base-b logarithm of the product \(MN\) can be found using the equation: \(\log_b (MN) = \log_b M + \log_b N\). This rule allows us to separate the logarithm of a product into the sum of the logarithms of its factors.
02

Applying the Product Rule

Let's use an example with numbers to make the application of the rule more concrete. Suppose \(b = 2\), \(M = 4\), and \(N = 8\). Then \(\log_2 4 = 2\) and \(\log_2 8 = 3\). According to the product rule, the logarithm base 2 of the product \(4*8 = 32\) can be calculated as the sum of \(\log_2 4\) and \(\log_2 8\): \(\log_2 32 = \log_2 4 + \log_2 8 = 2 + 3 = 5\). Hence, the logarithm base 2 of 32 is 5, therefore the rule is correctly applied.

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Most popular questions from this chapter

Find the domain of each logarithmic function. $$ f(x)=\log _{5}(x+4) $$

In parts (a)-(c), graph \(f\) and \(g\) in the same viewing rectangle. a. \(f(x)=\ln (3 x), g(x)=\ln 3+\ln x\) b. \(f(x)=\log \left(5 x^{2}\right), g(x)=\log 5+\log x^{2}\) c. \(f(x)=\ln \left(2 x^{3}\right), g(x)=\ln 2+\ln x^{3}\) d. Describe what you observe in parts (a)-(c). Generalize this observation by writing an equivalent expression for \(\log _{b}(M N),\) where \(M>0\) and \(N>0\) e. Complete this statement: The logarithm of a product is equal to _____.

Graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of \(f\). $$ f(x)=\ln x, g(x)=\ln x+3 $$

Evaluate or simplify each expression without using a calculator. $$ 10^{\log 53} $$

Evaluate or simplify each expression without using a calculator. $$ e^{\ln 5 x^{2}} $$
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