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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 18
Chapter 5, Problem 18

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)

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Identify the function g(x) given as \(g(x) = \left( \frac{1}{4} \right)^x\).
To find \(g(-3)\), substitute \(x = -3\) into the function: \(g(-3) = \left( \frac{1}{4} \right)^{-3}\).
Recall the property of exponents: \(a^{-n} = \frac{1}{a^n}\). Apply this to rewrite \(\left( \frac{1}{4} \right)^{-3}\) as \$4^3$.
Calculate \$4^3$ by multiplying 4 by itself three times: \(4 \times 4 \times 4\).
After finding the value, round the result to the nearest thousandth if necessary.

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Key Concepts

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

Exponential Functions

Exponential functions have the form f(x) = a^x, where the base a is a positive constant. They model growth or decay processes and are evaluated by raising the base to the power of the input x. Understanding how to compute values for given x is essential.
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Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the positive exponent, i.e., a^(-x) = 1/(a^x). This concept is crucial when evaluating functions like g(x) = (1/4)^x at negative values of x.
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Rounding to a Specified Decimal Place

Rounding involves approximating a number to a certain number of decimal places, here to the nearest thousandth (three decimal places). This ensures answers are presented clearly and consistently, especially when dealing with irrational or long decimal values.
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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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Determine whether each function graphed or defined is one-to-one. y = 2x - 8

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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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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. log4 1/64 = -3

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