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Chapter 4: Problem 42

Evaluate each expression without using a calculator. $$7^{\log _{7} 23}$$

Short Answer

Expert verified
23

Step by step solution

01

- Understand the properties of logarithms

According to the logarithm rules, if the base 'b' of the power is equal to the base 'b' of the logarithm, the power of the 'b' is equal to the logarithm's argument. Therefore, \(b^{\log _{b} x} = x\).
02

- Apply the rule to the given problem

Regarding the given expression \(7^{\log _{7} 23}\), it can be observed that the base of the power (7) equals the base of the logarithm (also 7). According to the rule explained in step 1, this power of the seven will be equal to the argument of the logarithm, which is 23.

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Most popular questions from this chapter

Solve each equation. Check each proposed solution by direct substitution or with a graphing utility. $$ (\ln x)^{2}=\ln x^{2} $$

In Exercises \(125-128,\) determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \ln \sqrt{2}=\frac{\ln 2}{2} $$

Evaluate each expression without using a calculator. $$ \log _{5}\left(\log _{2} 32\right) $$

Logarithmic models are well suited to phenomena in which growth is initially rapid but then begins to level off. Describe something that is changing over time that can be modeled using a logarithmic function.

Evaluate or simplify each expression without using a calculator. $$ \ln e^{6} $$
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