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Chapter 4: Problem 144

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

Short Answer

Expert verified
The exact value is 0.

Step by step solution

01

Simplify the Innermost Logarithm

Start with the innermost logarithm \( \log _{2} 8 \). We know that 2 raised to the power of 3 is 8. So, \( \log _{2} 8 = 3 \). Then we substitute this value back into the whole equation, giving \( \log _{4}\left[\log _{3}\left(3\right)\right] \).
02

Simplify the Second Logarithm

Now we simplify the second logarithm \( \log _{3} 3 \). We know that 3 raised to the power of 1 is 3. So, \( \log _{3} 3 = 1 \). Then we substitute this value back into the whole equation, giving \( \log _{4} 1 \).
03

Simplify the Final Logarithm

Now we simplify the final logarithm \( \log _{4} 1 \). We know that 4 raised to the power of 0 is 1. So, \( \log _{4} 1 = 0 \).

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Most popular questions from this chapter

Find the domain of each logarithmic function. $$ f(x)=\log (7-x) $$

Find the domain of each logarithmic function. $$ f(x)=\log _{5}(x+6) $$

Graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of \(f\). $$ f(x)=\ln x, g(x)=\ln (x+3) $$

The bar graph indicates that the percentage of first-year college students expressing antifeminist views declined after \(1970 .\) Use this information to solve. (GRAPH CANNOT COPY). The function $$f(x)=-7.52 \ln x+53$$ models the percentage of first-year college men, \(f(x)\) expressing antifeminist views (by agreeing with the statement) \(x\) years after 1969. a. Use the function to find the percentage of first-year college men expressing antifeminist views in 2008 . Round to one decimal place. Does this function value overestimate or underestimate the percentage displayed by the graph? By how much? b. Use the function to project the percentage of first-year college men who will express antifeminist views in 2015 . Round to one decimal place.

Exercises \(134-136\) will help you prepare for the material covered in the next section. Solve: \(x(x-7)=3\)
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