Q. 44 Solve each exponential equation.... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 5: Q. 44 (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.
${(\frac{4}{3})}^{1 - x} = 5^{x}$

Short Answer

Expert verified

The exact solution is $x = \frac{\ln (\frac{4}{3})}{\ln (\frac{20}{3})}$ and the rounded solution is $x 鈮� 0.152$ .

Step by step solution

Step 1. Given information.

The given equation is:

${(\frac{4}{3})}^{1 - x} = 5^{x}$

Step 2. Concepts and rules.

The following rules are required to solve the given equation.

If $\log_{a} M = \log_{a} N$ , then $M = N$ .

$\begin{array}{rcl} \ln_{a} M^{r} & = & r \ln M \\ \ln M + \ln N & = & \ln (M N) \end{array}$

Step 3. Determine the solution.

Taking ln on both sides, we get

$\begin{array}{rcl} \ln {(\frac{4}{3})}^{1 - x} & = & \ln 5^{x} \\ (1 - x) \ln (\frac{4}{3}) & = & x \ln 5 \\ \ln (\frac{4}{3}) - x \ln (\frac{4}{3}) & = & x \ln 5 \\ \ln (\frac{4}{3}) & = & x \ln 5 + x \ln (\frac{4}{3}) \end{array}$

Isolate the variable.

$\begin{array}{rcl} \ln (\frac{4}{3}) & = & x (\ln 5 + \ln (\frac{4}{3})) \\ \ln (\frac{4}{3}) & = & x \ln (\frac{20}{3}) \\ \frac{\ln (\frac{4}{3})}{\ln (\frac{20}{3})} & = & x \\ x & 鈮� & 0.152 \end{array}$

The exact solution is $x = \frac{\ln (\frac{4}{3})}{\ln (\frac{20}{3})}$ and the rounded solution is $x 鈮� 0.152$ .

Step 4. Verify the result by the graphing utility.

Plot the graphs of $y_{1} = {(\frac{4}{3})}^{1 - x}$ and $y_{2} = 5^{x}$ on a coordinate plane. The intersection point of these curves is the solution of the given equation.

The graphs intersect each other at $x 鈮� 0.152$ . Hence, the solution is verified.

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91影视

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.
${(\frac{4}{3})}^{1 - x} = 5^{x}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Concepts and rules.

Step 3. Determine the solution.

Step 4. Verify the result by the graphing utility.

One App. One Place for Learning.

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91影视

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.431-x=5x

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Concepts and rules.

Step 3. Determine the solution.

Step 4. Verify the result by the graphing utility.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Theoretical and Mathematical Physics

Mechanics Maths

Pure Maths

Applied Mathematics

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.
${(\frac{4}{3})}^{1 - x} = 5^{x}$