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Chapter 12: Problem 29

Evaluate each expression without using a calculator. $$\log _{7} \sqrt{7}$$

Short Answer

Expert verified
Therefore, \(\log_{7}\sqrt{7} = 1/2\).

Step by step solution

01

Convert the Root to Fractional Exponent

Represent the square root of 7 as a fractional exponent. According to the rule, the square root can be written as an exponent of 1/2. So, \( \sqrt{7} = 7^{1/2} \).
02

Apply Logarithm Property

Next, use the property of logarithms that \(log_b(a^m)=m*log_b(a)\) to break down the expression. So, we have \(log_{7}(7^{1/2})=(1/2)*log_{7}(7)\).
03

Simplify the terms

We know that \(log_{b}(b)=1\) because any number raised to power 1 gives itself. So, \(log_{7}(7)=1\). Substituting this in the equation, we get \((1/2)*log_{7}(7) = (1/2)*1 = 1/2 \).

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Most popular questions from this chapter

Without using a calculator, find the exact value of \(\log _{4}\left[\log _{3}\left(\log _{2} 8\right)\right]\)

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