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Chapter 13: Problem 23

Write as the sum or difference of logarithms and simplify, if possible. Assume all variables represent positive real numbers. $$\log _{2} \frac{8}{k}$$

Short Answer

Expert verified
The given logarithm expression can be rewritten as a sum or difference of logarithms using the quotient rule: \(\log _{2} \frac{8}{k} = \log_2{8} - \log_2{k}\). After simplifying the expression, we get the final result: \(3 - \log_2{k}\).

Step by step solution

01

Apply the Quotient Rule for Logarithms

Recall the quotient rule for logarithms states that \(\log_b{(\frac{M}{N})} = \log_b{M} - \log_b{N}\). Applying this rule, we have: \(\log _{2} \frac{8}{k} = \log_2{8} - \log_2{k}\)
02

Simplify the Logarithm Expression

Now, we know that \(2^3 = 8\), so the logarithm with the base 2 of 8 is 3, because \(2^3 = 8\). Therefore, the expression simplifies to: \(\log_2{8} - \log_2{k} = 3 - \log_2{k}\) The final expression as a sum or difference of logarithms is: \(3 - \log_2{k}\)

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