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Chapter 13: Problem 45

Write as the sum or difference of logarithms and simplify, if possible. Assume all variables represent positive real numbers. $$\log _{5} \sqrt{5 c}$$

Short Answer

Expert verified
\(\frac{1}{2} + \frac{1}{2}\log _{5} c\)

Step by step solution

01

Recall properties of logarithms

Recall the following properties of logarithms: 1. Power rule: \(\log _b(x^y) = y\log _b x\) 2. Product rule: \(\log _b(x*y) = \log _b x + \log _b y\) We will use these properties to simplify the given logarithmic expression.
02

Apply the power rule

Apply the power rule to the expression. The square root can be written as a power of \(\frac{1}{2}\): \(\log _{5} \sqrt{5 c} = \log _{5} {(5c)}^{\frac{1}{2}}\) By applying the power rule, we get: \(\frac{1}{2}\log _{5} (5c)\)
03

Apply the product rule

Now we will apply the product rule to the expression inside the logarithm: \(\frac{1}{2}\log _{5} (5c) = \frac{1}{2}(\log _{5} 5 + \log _{5} c)\)
04

Simplify

Recall that \(\log _{5} 5 = 1\), so our expression simplifies to: \(\frac{1}{2}(1 + \log _{5} c) = \frac{1}{2} + \frac{1}{2}\log _{5} c\) Therefore, the given expression \(\log _{5} \sqrt{5 c}\) can be written as the sum of logarithms and simplified as: \[\frac{1}{2} + \frac{1}{2}\log _{5} c\]

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