Chapter 5: Q. 40 (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
$0.3 (4^{0.2 x}) = 0.2$

Short Answer

Expert verified

$x = 5. \log_{4} \frac{2}{3} = 5 \cdot \frac{\ln \frac{2}{3}}{\ln 4} \approx - 1.462$

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.

$0.3 (4^{0.2 x}) = 0.2$

Step 2. Explanation

$\begin{array}{l} {Wecansolveusingformulalog}_{a} a^{x} = x \\ Herea=4 \\ Soweget \\ \Rightarrow 4^{0.2 x} = \frac{2}{3} \\ \Rightarrow \log_{4} 4^{0.2 x} = \log_{4} \frac{2}{3} \\ \Rightarrow 0.2 x = \log_{4} \frac{2}{3} \\ \Rightarrow \frac{2}{10} x = \log_{4} \frac{2}{3} \\ \Rightarrow \frac{1}{5} x = \log_{4} \frac{2}{3} \\ \Rightarrow x = 5. \log_{4} \frac{2}{3} \\ {Nowusingformulalog}_{a} M = \frac{\log_{a} M}{\log_{b} a} \\ Forbwewilltakeb=e \\ So, \\ x = 5. \log_{4} \frac{2}{3} = 5 \cdot \frac{\ln \frac{2}{3}}{\ln 4} \approx - 1.462 \end{array}$

Step 3. Verify the solution with the graph.

The graph of the given function is:

Here, we can see that the graph is a straight line passes through the point $x = - 1.462$ .

Thus it is verified that the solution is correct.

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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
$0.3 (4^{0.2 x}) = 0.2$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

Step 3. Verify the solution with the graph.

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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 utility0.3(40.2x)=0.2

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

Step 3. Verify the solution with the graph.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Calculus

Theoretical and Mathematical Physics

Discrete Mathematics

Statistics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

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
$0.3 (4^{0.2 x}) = 0.2$