Textbook Question
Use a graphing utility and the change-of-base property to graph each function. y = log3 x
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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 81
Solve each logarithmic equation in Exercises 49–92. 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. 2 log x=log 25
Verified step by step guidance1
Start with the given equation: \(2 \log x = \log 25\).
Use the logarithmic property that allows you to move the coefficient in front of the log as an exponent inside the log: \(2 \log x = \log x^{2}\), so rewrite the equation as \(\log x^{2} = \log 25\).
Since the logarithms on both sides have the same base (assumed to be 10), set the arguments equal to each other: \(x^{2} = 25\).
Solve the equation \(x^{2} = 25\) by taking the square root of both sides, remembering to consider both positive and negative roots: \(x = \pm 5\).
Check the domain of the original logarithmic expression: since \(\log x\) is defined only for \(x > 0\), reject \(x = -5\) and keep \(x = 5\) as the valid solution.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Properties of Logarithms
Understanding the properties of logarithms, such as the product, quotient, and power rules, is essential. In this problem, the power rule allows rewriting 2 log x as log(x^2), enabling simplification and comparison of logarithmic expressions.
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Change of Base Property
Domain of Logarithmic Functions
Logarithmic functions are only defined for positive arguments. When solving equations like 2 log x = log 25, it is crucial to ensure that the solution for x keeps the argument of all logarithms positive, rejecting any values that do not satisfy this domain restriction.
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5:26Graphs of Logarithmic Functions
Solving Logarithmic Equations
Solving logarithmic equations often involves rewriting the equation using logarithm properties, then converting to exponential form to isolate the variable. After finding exact solutions, decimal approximations can be calculated when necessary.
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5:02Solving Logarithmic Equations
Related Practice
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Use a graphing utility and the change-of-base property to graph each function. y = log2 (x + 2)
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Textbook Question
Evaluate or simplify each expression without using a calculator. log 100
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Textbook Question
Expand:
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Textbook Question
Solve each logarithmic equation in Exercises 49–92. 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(3x−3)=log(x+1)+log 4
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Textbook Question
Evaluate or simplify each expression without using a calculator. log 107
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