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Chapter 4: Problem 29

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

Short Answer

Expert verified
The value of \( \log _{7} \sqrt{7} \) is \(\frac{1}{2} \)

Step by step solution

01

Understand the Problem

Here, we need to evaluate the expression \( \log _{7} \sqrt{7} \) without using a calculator. The base of the logarithm is 7, and its argument is \(\sqrt{7}\), which is the same as \(7^{1/2}\). So, we are essentially trying to find the exponent that would get us from 7 (the base) to \( \sqrt{7} \) (the argument). The understanding of properties of logarithm is vital in carrying out this step.
02

Apply the rule of logarithm

Having seen that our argument is a power, we can apply the rule of logarithm that allows us to move this exponent out front. In math terms, this can be described as: \( \log_b (a^n) = n \log_b (a) \). Hence, we can step down our expression \( \log _{7} \sqrt{7} \) to \( \frac{1}{2} \log _{7} 7 \) by bringing down the exponent 1/2, which is the square root.
03

Evaluate the logarithm

The logarithm \( \log_{7} 7 \) implies a question: to what power should we raise 7 to obtain 7? Since anything raised to the power of 1 is itself, \(\log_{7} 7\) is equal to 1. Definitions like this are true for any logarithm, base and its argument being the same - it's always equal to 1. So our expression now simplifies to \( \frac{1}{2} \times 1 = \frac{1}{2} \)

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Most popular questions from this chapter

The logistic growth function $$ P(x)=\frac{90}{1+271 e^{-0.122 x}} $$ models the percentage, \(P(x),\) of Americans who are \(x\) years old with some coronary heart disease. Use the function to solve Exercises \(43-46\) What percentage of 20 -year-olds have some coronary heart disease?

Evaluate the indicated logarithmic expressions without using a calculator. a. Evaluate: \(\log _{2} 32\) b. Evaluate: \(\log _{2} 8+\log _{2} 4\) c. What can you conclude about \(\log _{2} 32,\) or \(\log _{2}(8 \cdot 4) ?\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because the equations $$\log (3 x+1)=5 \text { and } \log (3 x+1)=\log 5$$ are similar, I solved them using the same method.

The percentage of adult height attained by a girl who is \(x\) years old can be modeled by $$f(x)=62+35 \log (x-4)$$ where \(x\) represents the girl's age (from 5 to 15 ) and \(f(x)\) represents the percentage of her adult height. Round answers to the nearest tenth of a percent. Approximately what percentage of her adult height has a girl attained at age \(13 ?\)

Make Sense? In Exercises \(73-76\), determine whether each statement makes sense or does not make sense, and explain your reasoning. Because carbon-14 decays exponentially, carbon dating can determine the ages of ancient fossils.
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