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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 equation \[ \log(a*b) = \log(a) + \log(b) \] derived from the verbal statement is true. This is a fundamental property of logarithms known as the product rule.

Step by step solution

01

Translating Verbal Statement to Equation

The verbal statement 'The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers' can be translated into an equivalent mathematical equation as: \[ \log(a*b) = \log(a) + \log(b) \]
02

Checking If the Equation is True or False

This equation is one of the basic logarithmic properties, also known as the product rule for logarithms, and always holds true for any value of \(a\) and \(b\) that is in the domain of the logarithm function.

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Most popular questions from this chapter

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$6 e^{1-x}=25$$

Determine whether the statement is true or false. Justify your answer. The domain of a logistic growth function cannot be the set of real numbers.

The demand equation for a smart phone is \(p=5000\left(1-\frac{4}{4+e^{-0.002 x}}\right)\) Find the demand \(x\) for a price of \((\mathrm{a}) p=\$ 169\) and (b) \(p=\$ 299\)

Use a graphing utility to graph the functions \(y_{1}=\ln x-\ln (x-3)\) and \(y_{2}=\ln \frac{x}{x-3}\) in the same viewing window. Does the graphing utility show the functions with the same domain? If so, should it? Explain your reasoning.

Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. $$h(x)=\ln (x+5)$$
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