/*! 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 41 Solve each logarithmic equation.... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 41

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log _{3} x=4$$

Short Answer

Expert verified
The exact value solution for the equation \(\log_{3} x = 4\) is \(x = 81\).

Step by step solution

01

Understand the properties of the logarithm in the equation

The equation is in logarithmic form with base 3. Logarithms and exponentials are inverses of each other, meaning that they can be rewritten in terms of each other. So, the equation can be written in exponential form as \(3^4 = x\).
02

Calculate the exponential expression

Calculate \(3^4\), which involves raising 3 to the power of 4. This gives \(3*3*3*3 = 81\).
03

Check the domain

Since the input of a logarithm (x in this case) should be greater than zero, the domain of the logarithmic function is \(x > 0\). In this case, since 81 is greater than 0, the solution is valid.

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

Find \(\ln 2\) using a calculator. Then calculate each of the following: \(1-\frac{1}{2} ; \quad 1-\frac{1}{2}+\frac{1}{3} ; \quad 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}\) \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}, \ldots\) Describe what you observe.

Complete the table for a savings account subject to continuous compounding ( \(A=P e^{n}\) ). Round answers to one decimal place. $$\begin{array}{l|c|l|c} \hline \begin{array}{l} \text { Amount } \\ \text { Invested } \end{array} & \begin{array}{l} \text { Annual Interest } \\ \text { Rate } \end{array} & \begin{array}{l} \text { Accumulated } \\ \text { Amount } \end{array} & \begin{array}{l} \text { Time } t \\ \text { in Years } \end{array} \\ \hline \$ 17,425 & 4.25 \% & \$ 25,000 & \\ \hline \end{array}$$

Explain how to solve an exponential equation when both sides can be written as a power of the same base.

The function \(P(t)=145 e^{-0.092 t}\) models a runner's pulse, \(P(t),\) in beats per minute, \(t\) minutes after a race, where \(0 \leq t \leq 15 .\) Graph the function using a graphing utility. [TRACE] along the graph and determine after how many minutes the runner's pulse will be 70 beats per minute. Round to the nearest tenth of a minute. Verify your observation algebraically.

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$5^{x}=3 x+4$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

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.