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Chapter 3: Problem 49

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $$\log _{5} \frac{5}{x}$$

Short Answer

Expert verified
The expanded form of the given logarithmic expression \(\log _{5} \frac{5}{x}\) is \(1 - \log_5 x\).

Step by step solution

01

Identification of Logarithmic Form

Firstly, identify the form of the given logarithm. The task has given the logarithmic expression \(\log _{5} \frac{5}{x}\). This is in the form of a quotient (a division) inside a logarithm.
02

Applying the Quotient Rule

The quotient rule in logarithms states that \(\log_{b}(a / c)=\log_{b} a - \log_{b} c\). Apply the quotient rule to the given logarithm to separate the numerator and the denominator: \(\log _{5} \frac{5}{x}= \log_5 5 - \log_5 x\)
03

Simplifying the Expressions

Now simplify the first term. Any log base \(b\) of \(b\) is simply 1. So, \(\log_5 5 = 1\). Also, since there is no operation to perform on the second term, leave it as \(\log_5 x\). So, the final simplified expression is: \(1 - \log_5 x\)

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