Chapter 5: Q. 39 (page 315)

Solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places. Verify your results using a graphing utility.
$5 (2^{3 x}) = 8$

Short Answer

Expert verified

$x = \frac{1}{3} . \log_{2} \frac{8}{5} = \frac{1}{3} \cdot \frac{\ln \frac{8}{5}}{3 \ln 2} \approx 0.226$

Step by step solution

Step 1. Given information

Solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places. Verify your results using a graphing utility.

$5 (2^{3 x}) = 8$

Step 2. Explanation

$\begin{array}{l} {Wecansolveusingformulalog}_{a} a^{x} = x \\ Herea=2 \\ Soweget \\ \Rightarrow 2^{3 x} = \frac{8}{5} \\ \Rightarrow \log_{2} 2^{3 x} = \log_{2} \frac{8}{5} \\ \Rightarrow 3 x = \log_{2} \frac{8}{5} \\ \Rightarrow x = \frac{1}{3} . \log_{2} \frac{8}{5} \\ {Nowusingformulalog}_{a} M = \frac{\log_{a} M}{\log_{b} a} \\ Forbwewilltakeb=e \\ So, \\ x = \frac{1}{3} . \log_{2} \frac{8}{5} = \frac{1}{3} \cdot \frac{\ln \frac{8}{5}}{3 \ln 2} \approx 0.226 \end{array}$

Step 3. Verify the result

Plot the graph of the given equation as shown below

Hence,

The solution to the equation is $x = 0.226$ .

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Solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places. Verify your results using a graphing utility.
$5 (2^{3 x}) = 8$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

Step 3. Verify the result

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Solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places. Verify your results using a graphing utility.5(23x)=8

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

Step 3. Verify the result

One App. One Place for Learning.

Most popular questions from this chapter

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Solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places. Verify your results using a graphing utility.
$5 (2^{3 x}) = 8$