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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 5: Q. 45 (page 349)

Solve each equation. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places. Verify your results using a graphing utility: $9^{x} + 4 脳 3^{x} - 3 = 0$

Short Answer

Expert verified

The solution set of $x = \{- 0.398\}$

Step by step solution

01

Step 1. Given information

An equation, $9^{x} + 4 脳 3^{x} - 3 = 0$

02

Step 2. Required to find

Solve the given equation to find x.

03

Step 3. Solving the equation

Given, $9^{x} + 4 脳 3^{x} - 3 = 0$

${(3^{x})}^{2} + 4 脳 3^{x} - 3 = 0$

Put, $3^{x} = y$

$鈬� y^{2} + 4 y - 3 = 0$

For quadratic equation, role="math" localid="1646580117285" $a y^{2} + b y + c = 0$ , using quadratic formula, $y = \frac{- b 卤 \sqrt{b^{2} - 4 a c}}{2 a}$

$鈬� y = \frac{- 4 卤 \sqrt{{(4)}^{2} - 4 (1) (- 3)}}{2 (1)}$

$鈬� y = \frac{- 4 卤 \sqrt{16 + 12}}{2}$

$鈬� y = \frac{- 4 卤 \sqrt{28}}{2}$

$鈬� y = \frac{- 4 卤 \sqrt{4 脳 7}}{2}$

$鈬� y = \frac{2 (- 2 卤 \sqrt{7})}{2}$

$鈬� y = - 2 - \sqrt{7}, y = - 2 + \sqrt{7}$

Only $y = - 2 + \sqrt{7}$ is the solution as $3^{x}$ is positive.

$鈬� 3^{x} = - 2 + \sqrt{7}$

Taking log on both sides,

$\ln 3^{x} = \ln (- 2 + \sqrt{7})$

$鈬� x \ln 3 = \ln (- 2 + \sqrt{7})$

role="math" localid="1646580844137" $鈬� x = \frac{\ln 0.645}{\ln 3}$

role="math" localid="1646580874376" $鈬� x = \frac{- 0.437}{1.098}$

$鈬� x = - 0.398$

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Most popular questions from this chapter

In Problems 61鈥�72, the function f is one-to-one. Find its inverse and check your answer.

$f (x) = \frac{3 x + 4}{2 x - 3}$

As the base a of an exponential function $f (x) = a^{x}$ , where $a > 1$ increases, what happens to the behavior of its graph for $x > o ?$ What happens to the behavior of its graph for $x < 0 ?$

Solve the equation $3^{x^{3}} = 9^{x}$

and verify your results using a graphing utility.

If $f (x) = a^{x}$ , show that $f (A + B) = f (A) . f (B)$ .

In Problems 61鈥�72, the function f is one-to-one. Find its inverse and check your answer.

$f (x) = \frac{4}{2 - x}$

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