Chapter 5: Q. 60 (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.
$3^{x} - 14 \cdot 3^{- x} = 5$

Short Answer

Expert verified

The exact solution is $x = \log_{3} 7$ and the rounded solution is $x \approx 1.771$ .

Step by step solution

Step 1. Rewrite the given equation.

The given equation is:

$3^{x} - 14 \cdot 3^{- x} = 5$

It can be rewritten as:

$3^{x} - \frac{14}{3^{x}} = 5$

Multiply both sides by $3^{x}$ .

$\begin{array}{rcl} (3^{x})^{2} - 14 & = & 5 (3^{x}) \\ (3^{x})^{2} - 5 (3^{x}) - 14 & = & 0 \end{array}$

Step 2. Substitute another variable and solve for it.

Let $3^{x} = u$ .

$\begin{array}{rcl} u^{2} - 5 u - 14 & = & 0 \\ u^{2} - 7 u + 2 u - 14 & = & 0 \\ u (u - 7) + 2 (u - 7) & = & 0 \\ (u + 2) (u - 7) & = & 0 \end{array}$

Use the zero product property.

$\begin{array}{rcl} u + 2 & = & 0 or u - 7 = 0 \\ u & = & - 2 or u = 7 \end{array}$

Step 3. Determine the solution.

Substitute $u = 3^{x}$ in the above solutions.

$3^{x} = - 2 or 3^{x} = 7$

The equation $3^{x} = - 2$ has no solution because $3^{x} > 0$ . Now, solve $3^{x} = 7$ .

localid="1648449916872" $\begin{array}{rcl} \ln 3^{x} & = & \ln 7 \\ x \ln 3 & = & \ln 7 \\ x & = & \frac{\ln 7}{\ln 3} \\ x & = & \log_{3} 7 \\ x & \approx & 1.771 \end{array}$

The exact solution is $x = \log_{3} 7$ and the rounded solution is $x \approx 1.771$ .

Step 4. Verify the result by the graphing utility.

Plot the graphs of $y_{1} = 3^{x} - 14 \cdot 3^{- x}$ and $y_{2} = 5$ on a coordinate plane. The x-coordinate of the intersection point is the solution of the given equation.

The graphs intersect each other at $x \approx 1.771$ . Hence, the solution is verified.

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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.
$3^{x} - 14 \cdot 3^{- x} = 5$

Short Answer

Step by step solution

Step 1. Rewrite the given equation.

Step 2. Substitute another variable and solve for it.

Step 3. Determine the solution.

Step 4. Verify the result by the graphing utility.

One App. One Place for Learning.

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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.3x-14·3-x=5

Short Answer

Step by step solution

Step 1. Rewrite the given equation.

Step 2. Substitute another variable and solve for it.

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

Mechanics Maths

Pure Maths

Probability and Statistics

Statistics

Applied Mathematics

Calculus

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.
$3^{x} - 14 \cdot 3^{- x} = 5$