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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 17
Chapter 5, Problem 17

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. 4(x-1) = 32x

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Start with the given equation: \(4^{x-1} = 3^{2x}\).
Rewrite the bases as powers of prime factors if possible. Note that \(4\) can be written as \$2^2$, so rewrite the equation as: \(\left(2^2\right)^{x-1} = 3^{2x}\).
Use the power of a power property: \(\left(a^m\right)^n = a^{mn}\), to simplify the left side: \(2^{2(x-1)} = 3^{2x}\).
Take the natural logarithm (or log base 10) of both sides to bring down the exponents: \(\ln\left(2^{2(x-1)}\right) = \ln\left(3^{2x}\right)\).
Apply the logarithm power rule: \(\ln(a^b) = b \ln(a)\), to get: \(2(x-1) \ln(2) = 2x \ln(3)\). Then solve this linear equation for \(x\) by expanding and isolating \(x\).

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Key Concepts

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

Exponential Equations

Exponential equations involve variables in the exponent position, such as 4^(x-1) = 3^(2x). Solving these requires understanding how to manipulate and equate expressions with different bases, often by applying logarithms or rewriting terms with common bases.
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Solving Exponential Equations Using Logs

Logarithms and Their Properties

Logarithms are the inverse operations of exponentials and are essential for solving equations where the variable is an exponent. Using properties like log(a^b) = b*log(a) allows us to bring down exponents and solve for the variable algebraically.
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Rounding and Exact vs. Approximate Solutions

Some solutions to equations are irrational and cannot be expressed exactly as fractions or decimals. Understanding when to provide exact forms (like logarithmic expressions) versus decimal approximations rounded to a specific place value is important for clear and accurate answers.
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Related Practice
Textbook Question

If the statement is in exponential form, write it in an equivalent logarithmic form. If the statement is in logarithmic form, write it in exponential form. log381=8\(\log\)_{\(\sqrt\)3}81=8

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Textbook Question

Determine whether each function graphed or defined is one-to-one. y = 2x - 8

Textbook Question

Find each value. If applicable, give an approximation to four decimal places. log 0.0022

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

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(2)

Textbook Question

If the statement is in exponential form, write it in an equivalent logarithmic form. If the statement is in logarithmic form, write it in exponential form. log5 5 = 1