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Chapter 13: Problem 34

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

Short Answer

Expert verified
The simplified expression for \(\log _{2} \sqrt{8}\) is \(\frac{3}{2}\).

Step by step solution

01

Simplify the square root

First, we need to rewrite the square root as an exponent. Recall that the square root of a number is the same as raising the number to the power of \(1/2\). So, we can rewrite the expression as: \(\log _{2} (8^{1/2})\)
02

Apply the power rule of logarithms

Now, we apply the power rule for logarithms which states that \(\log _b (x^y ) = y \log _b (x)\). Applying this rule to our expression gives: \((1/2) \log _{2} 8\)
03

Evaluate the logarithm

Now, we evaluate the logarithm \(\log _{2} 8\). Since \(2^3 = 8\), we have \(\log _{2} 8 = 3\). Finally, we substitute this value back into the expression: \((1/2) \log _{2} 8 = (1/2) \cdot 3\)
04

Simplify the expression

We now simplify the expression by multiplying the fraction with the logarithm value: \((1/2) \cdot 3 = 3/2\) As a result, the simplified expression is: $$\log _{2} \sqrt{8} = \frac{3}{2}$$

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