Textbook Question
Solve each equation. ln 3−ln(x+5)−ln x=0
Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 5, Problem 101
Write each equation in its equivalent exponential form. Then solve for x. log3 (x-1) = 2
Verified step by step guidance1
Recall the definition of logarithm: if \( \log_b a = c \), then the equivalent exponential form is \( b^c = a \).
Apply this definition to the given equation \( \log_3 (x-1) = 2 \). Rewrite it as \( 3^2 = x - 1 \).
Calculate the exponential expression \( 3^2 \) (you can leave it as \( 3^2 \) for now if you prefer).
Set up the equation \( 3^2 = x - 1 \) and solve for \( x \) by adding 1 to both sides: \( x = 3^2 + 1 \).
Check the solution by substituting \( x \) back into the original logarithmic equation to ensure the argument \( x - 1 \) is positive and the equation holds true.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition of Logarithms
A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log_b(a) = c means b^c = a. Understanding this definition is essential to rewrite logarithmic equations in exponential form.
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Logarithms Introduction
Converting Logarithmic Equations to Exponential Form
To solve logarithmic equations, rewrite them in exponential form using the equivalence log_b(a) = c ⇔ b^c = a. This conversion simplifies solving for the variable inside the logarithm by turning the equation into an exponential equation.
Recommended video:
5:02Solving Logarithmic Equations
Solving Exponential Equations
Once the equation is in exponential form, solve for the variable by isolating it. This may involve basic algebraic operations such as addition, subtraction, multiplication, division, or taking roots, depending on the equation's complexity.
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5:47Solving Exponential Equations Using Logs
Related Practice
Textbook Question
Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. log3 (7) = 1/[log7 (3)]
Textbook Question
In Exercises 105–108, evaluate each expression without using a calculator. log5 (log2 32)
Textbook Question
Solve each equation.
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Textbook Question
In Exercises 105–108, evaluate each expression without using a calculator. log5 (log7 7)
Textbook Question
In Exercises 101–104, write each equation in its equivalent exponential form. Then solve for x. log4x=-3