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Chapter 13: Problem 24

Write as the sum or difference of logarithms and simplify, if possible. Assume all variables represent positive real numbers. $$\log _{3} \frac{x}{9}$$

Short Answer

Expert verified
The short answer is: \(\log_{3}\frac{x}{9} = \log_{3}(x) - 2\).

Step by step solution

01

Apply the properties of logarithms

We will apply the following logarithm properties: 1. Division becomes a subtraction: \(\log_b (a/b) = \log_b(a) - \log_b(b)\) Using property 1: \(\log_{3}\frac{x}{9} = \log_{3}(x) - \log_{3}(9)\)
02

Simplify the expression

Now, we can simplify the second term of the expression using the property: 2. If \(\log_b a^c = c\), then \(\log_b (a^p) = p\) Since \(3^2 = 9\), we can write 9 as \(3^2\): \(\log_{3}(x) - \log_{3}(3^2) = \log_{3}(x) - 2\) So, our final expression is: \(\log_{3}\frac{x}{9} = \log_{3}(x) - 2\)

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