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Chapter 3: Problem 142

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

Short Answer

Expert verified
The statement is true. The equation derived from the verbal statement \[ \log(\frac{a}{b}) = \log(a) - \log(b) \] is valid in accordance with the standard identities of logarithms, assuming that a and b are positive real numbers and b is not zero.

Step by step solution

01

Formulate an Equation

According to the verbal statement, we can translate it into an equation as follows: \[ \log(\frac{a}{b}) = \log(a) - \log(b) \], where a and b are the two numbers with b 鈮� 0 to avoid division by zero.
02

Evaluate the Statement

This equation represents one of the fundamental identities of logarithms. If the base of the logarithms is the same for every term, this statement would generally hold true. Therefore, the verbal statement is true.
03

Justify the Discussion

The the quote from the verbal statement - 'The logarithm of the quotient of two numbers is equal to the difference of the logarithms of the numbers.' holds up under the logarithmic laws and identities. The equation presented is suitable for any two numbers a and b, given that a and b are greater than 0, and b is not equal to 0.

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Most popular questions from this chapter

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$e^{-2 x}-2 x e^{-2 x}=0$$

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