/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 38 Solve the exponential equation a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 38

Solve the exponential equation algebraically. Approximate the result to three decimal places. $$3^{x^{2}}=7^{6-x}$$

Short Answer

Expert verified
Solving the given equation will give us two valid values for x. Those values, rounded to three decimal places, will be the solutions to the exponential equation.

Step by step solution

01

Application of Natural Logarithm

Apply the natural logarithm (ln) to both sides of the equation to simplify it. This results in \(ln(3^{x^{2}}) = ln(7^{6-x})\). According to the property of logarithms, we can pull down the exponents in front, which gives us \(x^{2} \cdot ln(3) = (6-x) \cdot ln(7)\).
02

Rearranging the Equation

Rearrange the equation so that terms with x are on one side and constants on the other side. This results in \(x^{2} \cdot ln(3) + x \cdot ln(7) - 6 \cdot ln(7) = 0\).
03

Transforming the Equation into Quadratic Form

Treat \(ln(3)\) as 'a' and \(ln(7)\) as 'b' (both of which are constants), to transform the equation into a standard quadratic equation of the form \(ax^{2} + bx + c = 0\). This gives us \(x^{2} \cdot a + x \cdot b - 6 \cdot b = 0\).
04

Solving the Quadratic Equation

Afterwards, solve the quadratic equation. It can be solved either by factoring, completing the square or the quadratic formula. The quadratic formula, \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\), will be used in this case. Insert the values of a, b, and c and solve for x.
05

Approximation

Finally, approximate the values of x to three decimal places as mentioned in the problem statement.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Writing a Natural Logarithmic Equation In Exercises \(53-56,\) write the exponential equation in logarithmic form. $$e^{1 / 2}=1.6487 \ldots$$

Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. $$h(x)=\ln (x+5)$$

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$\ln (x+1)=2-\ln x$$

Write the logarithmic equation in exponential form. $$\ln 7=1.945 \ldots$$

Function \(\quad\) Value $$ g(x)=-\ln x \quad x=\frac{1}{2}$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.