/*! 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 119 Solve the equation. \((\log x)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 119

Solve the equation. \((\log x)^{2}=\log x^{2}\)

Short Answer

Expert verified
x = 1 or x = 100.

Step by step solution

01

Simplify the right-hand side

Rewrite \(\text{log}\thinspace(x^2)\) using logarithmic properties. The property \(\text{log}\thinspace(a^b) = b\thinspace\text{log}\thinspace(a)\) can be used here. Therefore, \(\text{log}\thinspace(x^2) = 2\thinspace\text{log}\thinspace(x)\).
02

Set up the simplified equation

Using the result from Step 1, the equation \((\text{log}\thinspace(x))^2=\text{log}\thinspace(x^2)\) becomes \((\text{log}\thinspace(x))^2 = 2\thinspace\text{log}\thinspace(x)\).
03

Let \(y = \text{log}\thinspace(x)\)

Introduce a new variable \(y\) where \(y = \text{log}\thinspace(x)\). Substitute this into the equation from Step 2, giving \(y^2 = 2y\).
04

Solve the quadratic equation

Rearrange the equation \(y^2 = 2y\) to form a standard quadratic equation: \(y^2 - 2y = 0\). Factorizing gives \(y(y - 2) = 0\). Thus, the solutions are \(y = 0\) or \(y = 2\).
05

Back-substitute to find \(x\)

Recall that \(y = \text{log}\thinspace(x)\). So the solutions for \(y\) give the equations \(\text{log}\thinspace(x) = 0\) and \(\text{log}\thinspace(x) = 2\).
06

Solve for \(x\)

Solve each equation separately: \(\text{log}\thinspace(x) = 0\) implies \(x = 10^0 = 1\), and \(\text{log}\thinspace(x) = 2\) implies \(x = 10^2 = 100\).
07

Verify the solutions

Verify that both \(x = 1\) and \(x = 100\) satisfy the original equation \((\text{log}\thinspace(x))^2=\text{log}\thinspace(x^2)\). Both values satisfy the simplified form \((\text{log}\thinspace(x))^2 = 2\thinspace\text{log}\thinspace(x)\).

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.

Quadratic Equations
Quadratic equations are equations that take the form of \[ax^2 + bx + c = 0\]. These equations are called 'quadratic' because 'quad' refers to the variable being squared. Solving quadratic equations involves finding the roots, or solutions, that satisfy the equation. In general, the methods to solve them include:
  • Factoring
  • Using the quadratic formula
  • Completing the square

In our problem, we first transformed the equation into a quadratic form. By letting \[y = \text{log}\thinspace(x)\], we transformed the original logarithmic equation into \[y^2 = 2y\], which can be rewritten as \[y^2 - 2y = 0\]. This is now a standard quadratic equation. Next, we used factoring to solve for \[y\], finding that \[y = 0\] and \[y = 2\] were solutions.
Logarithmic Properties
Logarithmic properties are essential tools for simplifying and solving logarithmic equations. One of the most important properties we used is the power rule: \[ \text{log}(a^b) = b \thinspace \text{log}(a)\]. This rule allows us to handle exponents within a log expression. For the given problem, it allowed us to transform \[ \text{log}(x^2)\] into \[ 2 \thinspace \text{log}(x)\], significantly simplifying the equation. Other key logarithmic properties include:
  • \text{log}(ab) = \text{log}(a) + \text{log}(b)
  • \text{log}(a/b) = \text{log}(a) - \text{log}(b)
  • \text{log}_a(a) = 1 \thinspace \text{and} \thinspace \text{log}_a(1) = 0

Understanding and applying these properties can make solving logarithmic equations much more manageable.
Variable Substitution
Variable substitution is a method used to simplify complex expressions by introducing a new variable. In this exercise, we substituted \[y\] for \[ \text{log}\thinspace(x)\]. The original equation, \[ (\text{log}\thinspace(x))^2 = \text{log}\thinspace(x^2)\], became \[ y^2 = 2y \thinspace\] by substitution. This step is crucial as it simplifies the equation's structure, allowing us to apply quadratic solving techniques.
Variable substitution is particularly helpful when you recognize patterns in the equation that resemble simpler forms, such as quadratic equations, making the solving process much more straightforward.
Factoring
Factoring is a method used to break down a polynomial into simpler components, called factors, that together multiply to give the original polynomial. In our problem, we needed to factor the quadratic equation \[ y^2 - 2y = 0\]. Factoring gives us \[ y(y - 2) = 0\], indicating that either \[ y = 0\] or \[ y - 2 = 0\]. Therefore, the solutions are \[ y = 0\] and \[ y = 2\].
The key steps to factoring include:
  • Identifying common factors
  • Using the difference of squares
  • Factoring trinomials

Factoring simplifies the equation-solving process and is particularly effective for quadratic equations. Once factored, extracting the solutions becomes straightforward.

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

Given the functions defined by \(f(x)=2 x-1\) and \(g(x)=\frac{x+1}{2}\), a. Graph \(y=f(x), y=g(x),\) and the line \(y=x .\) Does the graph suggest that \(f\) and \(g\) are inverses? Why? b. Enter the following functions into the graphing editor. ( $$\mathrm{Y}_{1}=2 x-1$$ \(\mathrm{Y}_{2}=(x+1) / 2\) \(\mathrm{Y}_{3}=\mathrm{Y}_{1}\left(\mathrm{Y}_{2}\right)\) \(\mathrm{Y}_{4}=\mathrm{Y}_{2}\left(\mathrm{Y}_{1}\right)\) c. Create a table of points showing \(Y_{3}\) and \(Y_{4}\) for several values of \(x\). (Hint: Use the right and left arrows to scroll through the table editor to show functions \(Y_{3}\) and \(Y_{4}\).) Does the table suggest that \(f\) and \(g\) are inverses? Why?

A function of the form \(P(t)=a b^{t}\) represents the population of the given country \(t\) years after January 1,2000 . a. Write an equivalent function using base \(e\); that is, write a function of the form \(P(t)=P_{0} e^{k t} .\) Also, determine the population of each country for the year 2000 . $$\begin{array}{|l|c|c|c|} \hline \text { Country } & P(t)=a b^{t} & P(t)=P_{0} e^{k t} & \begin{array}{c} \text { Population } \\ \text { in } 2000 \end{array} \\ \hline \text { Haiti } & P(t)=8.5(1.0158)^{t} & & \\ \hline \text { Sweden } & P(t)=9.0(1.0048)^{t} & & \\ \hline \end{array}$$ b. The population of the two given countries is very close for the year 2000 , but their growth rates are different. Determine the year during which the population of each country will reach 10.5 million. c. Haiti had fewer people in the year 2000 than Sweden. Why did Haiti reach a population of 10.5 million sooner?

Compare the graphs of the functions. $$ \mathrm{Y}_{1}=\ln \left(\frac{x}{2}\right) \quad \text { and } \quad \mathrm{Y}_{2}=\ln x-\ln 2 $$

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(1024=19^{x}+4\)

Determine if the statement is true or false. For each false statement, provide a counterexample. For example, \(\log (x+y) \neq \log x+\log y\) because \(\log (2+8) \neq \log 2+\log 8\) (the left side is 1 and the right side is approximately 1.204 ). $$ \log _{2}(7 y)+\log _{2} 1=\log _{2}(7 y) $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

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.