/*! 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 6 Solve each equation. $$\log _{3... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 6

Solve each equation. $$\log _{3}\left(x^{2}\right)=4$$

Short Answer

Expert verified
x = 9 or x = -9

Step by step solution

01

Convert logarithmic form to exponential form

Rewrite the logarithmic equation \(\text{log}_{3}(x^2) = 4\) in its exponential form: \[3^4 = x^2 \]
02

Calculate the exponential value

Compute \3^4\: \[3^4 = 81 \]
03

Solve for x

Set up the equation \x^2 = 81\ and take the square root of both sides to solve for \x\: \[ x = \pm \sqrt{81} \] \[ x = \pm 9 \]

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Form
Understanding the concept of logarithmic form is essential for solving logarithmic equations.

The logarithmic form of an equation is typically written as \log_{\text{base}}(\text{value}) = \text{exponent} \. This form shows the exponent to which the base must be raised to get the value inside the log.

For example, if you have \log_3(x^2) = 4\, it means 3 raised to what power equals \(x^2\). Here, the base is 3, the value is \(x^2\), and the exponent is 4.

Logarithms are particularly helpful in reversing the exponentiation process, which makes them powerful tools for solving various types of equations.
Exponential Form
The exponential form of an equation is a way of expressing the same relationship as in logarithmic form but differently.

In exponential form, you write the base raised to the exponent equals the value. Using our previous example, \log_3(x^2) = 4\ can be rewritten as: \3^4 = x^2\.

This indicates that when you raise 3 to the power of 4, you get \(x^2\). Solving it further leads us to \3^4 = 81\.

Converting between logarithmic and exponential forms provides a different perspective that can often make the problem easier to solve. It's an essential skill for tackling complex logarithmic and exponential equations.
Square Root
The square root is a fundamental concept in algebra and is crucial for solving equations where the variable is squared.

Taking the square root of both sides of an equation removes the square and isolates the variable. For example, in the equation \(x^2 = 81\), taking the square root of both sides gives us: \[ x = \pm \sqrt{81} \]

Since the square root of 81 is 9, we get: \ x = \pm 9 \.

This means that x can be both positive or negative 9, i.e., \(x = 9\) or \(x = -9\).

Understanding square roots enables you to solve quadratic equations effectively and is integral to higher-level mathematics.

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

Determine whether each equation is true or false. $$ \frac{\log _{2}(16)}{\log _{2}(4)}=\log _{2}(4) $$

Which of the following expressions is not equal to \(\log \left(5^{2 / 3}\right) ?\) Explain. a) \(\frac{2}{3} \log (5)\) b) \(\frac{\log (5)+\log (5)}{3}\) c) \((\log (5))^{2 / 3}\) d) \(\frac{1}{3} \log (25)\)

Reading and Writing. After reading this section, write out the answers to these questions. Use complete sentences. What is the product rule for logarithms?

Use the base-change formula to find each logarithm to four decimal places. $$\log _{3}(5)$$

Composition of inverses. Graph the functions \(y=\ln \left(e^{x}\right)\) and \(y=e^{\ln (x)} .\) Explain the similarities and differences between the graphs.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.