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Ch. 4 - Inverse, Exponential, and Logarithmic Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 4 - Inverse, Exponential, and Logarithmic FunctionsProblem 20
Chapter 5, Problem 20

Solve each equation. x = log3 1/81

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1
Recognize that the equation is given in logarithmic form: \(x = \log_{3} \frac{1}{81}\). This means \(x\) is the exponent to which the base 3 must be raised to get \(\frac{1}{81}\).
Rewrite the equation in its equivalent exponential form: \(3^{x} = \frac{1}{81}\).
Express \(\frac{1}{81}\) as a power of 3. Since \(81 = 3^{4}\), then \(\frac{1}{81} = 3^{-4}\).
Set the exponents equal to each other because the bases are the same: \(3^{x} = 3^{-4}\) implies \(x = -4\).
Conclude that the solution to the equation is \(x = -4\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Logarithms and Their Definition

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log₃(1/81) asks for the power to which 3 must be raised to get 1/81. Understanding this definition is essential to solving logarithmic equations.
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Properties of Exponents

Since logarithms are inverses of exponents, knowing how to manipulate exponents is crucial. Recognizing that 81 is 3 raised to the 4th power (3⁴ = 81) helps rewrite 1/81 as 3⁻⁴, simplifying the logarithmic expression.
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Solving Logarithmic Equations

To solve equations like x = log₃(1/81), rewrite the logarithmic form into exponential form: 3ˣ = 1/81. Then, express 1/81 as a power of 3 and solve for x by equating exponents, which leads to the solution.
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Related Practice
Textbook Question

Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. 6(x+1) = 4(2x-1)

Textbook Question

For ƒ(x) = 3x and g(x)= (1/4)x find each of the following. Round answers to the nearest thousandth as needed. ƒ(-5/2)

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Find each value. If applicable, give an approximation to four decimal places. log(387 ×\(\times\) 23)

Textbook Question

Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. ex2=100e^{x^2} = 100

Textbook Question

Determine whether each function graphed or defined is one-to-one. y = -√(100 - x2)

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Textbook Question

For ƒ(x) = 3x and g(x)= (1/4)x find each of the following. Round answers to the nearest thousandth as needed. g(3/2)