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Chapter 4: Problem 91

You overhear a student talking about a property of logarithms in which division becomes subtraction. Explain what the student means by this.

Short Answer

Expert verified
The statement the student made is referring to the Quotient Rule of logarithms, which states that division inside a log with the same base can be rewritten as subtraction. In formula terms, it's represented as \(\log_b (M/N) = \log_b M - \log_b N\).

Step by step solution

01

Identify the Quotient Rule of Logarithms

The Quotient Rule of logarithms states that the division of two numbers within a logarithm (with the same base) can be represented as the subtraction of two logarithmic expressions. The general equation for this property is \(\log_b (M/N) = \log_b M - \log_b N\). Here, 'b' is the base of the logarithm, 'M' is the numerator and 'N' is a denominator.
02

Apply This Rule to a Specific Example

Suppose we have \(\log_2 (8/2)\). Applying the quotient rule, we can rewrite it as \(\log_2 8 - \log_2 2\). Solving this would give us \(3 - 1 = 2\), which is equal to \(\log_2 (8/2)\). This demonstrates that the division operation inside a logarithm has been successfully converted to a subtraction operation.

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Most popular questions from this chapter

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{9}(9 x) $$

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \frac{1}{2}\left(\log _{5} x+\log _{5} y\right)-2 \log _{5}(x+1) $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{2} \sqrt[5]{\frac{x y^{4}}{16}} $$

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 8 \ln (x+9)-4 \ln x $$

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 4 \ln x+7 \ln y-3 \ln z $$
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