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Chapter 3: Problem 87

Rewrite each verbal statement as an equation. Then decide whether the statement is true or false. Justify your answer. The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers.

Short Answer

Expert verified
The statement is true. The equation representing the statement is \( \log_b(MN) = \log_b(M) + \log_b(N) \). The justification is based on the rules of exponents and commonly accepted laws of logarithms.

Step by step solution

01

Express the Statement as an Equation

The verbal statement can be translated into the mathematical form: \( \log_b(MN) = \log_b(M) + \log_b(N) \) for any base \(b\), and any positive numbers \( M, N \) which are not equal to 1. This rule is valid for any base of logarithm.
02

Evaluate the Truth of the Statement

The statement is actually a classical property of logarithms. Therefore, it is indeed a true statement. It's the fundamental log rule, it is proved mathematically valid and widely used in algebra and calculus.
03

Justification

This property of logarithms can be justified using the rules of exponents. If \( b^x = M \) and \( b^y = N \), then \( b^{x+y} = MN \). Taking log base \( b \) on both sides, we get \( x + y = \log_b(MN) \) which is equal to \( \log_b(M) + \log_b(N) \). Thus, the given statement is justified and proven to be true.

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Most popular questions from this chapter

Using Properties of Logarithms In Exercises \(21-36,\) find the exact value of the logarithmic expression without using a calculator. (If this is not possible, then state the reason.) $$\log _{4} 16^{2}$$

Condensing a Logarithmic Expression In Exercises \(67-82,\) condense the expression to the logarithm of a single quantity. $$\frac{1}{4} \log _{3} 5 x$$

Condensing a Logarithmic Expression In Exercises \(67-82,\) condense the expression to the logarithm of a single quantity. $$\log x-2 \log (x+1)$$

Curve Fitting In Exercises \(89-92,\) find a logarithmic equation that relates \(y\) and \(x .\) Explain the steps used to find the equation. $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} & {6} \\\ \hline y & {0.5} & {2.828} & {7.794} & {16} & {27.951} & {44.091} \\\ \hline\end{array} $$

Using Properties of Logarithms In Exercises \(21-36\) , find the exact value of the logarithmic expression without using a calculator. (If this is not possible, then state the reason.) 3 $$\ln e^{4}$$
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