Textbook Question
Find each value. If applicable, give an approximation to four decimal places. log 518 - log 342
Ch. 4 - Inverse, Exponential, and Logarithmic Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 5, Problem 25
For ƒ(x) = 3x and g(x)= (1/4)x find each of the following. Round answers to the nearest thousandth as needed. g(-1.68)
Verified step by step guidance1
Identify the function you need to evaluate: here, it is \( g(x) = \left( \frac{1}{4} \right)^x \).
Substitute the given value \( x = -1.68 \) into the function: \( g(-1.68) = \left( \frac{1}{4} \right)^{-1.68} \).
Recall the property of exponents for negative powers: \( a^{-b} = \frac{1}{a^b} \). This means \( \left( \frac{1}{4} \right)^{-1.68} = 4^{1.68} \).
Calculate \( 4^{1.68} \) using a calculator or exponentiation method, but do not round yet.
Finally, round the result to the nearest thousandth as instructed.
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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 for solving problems involving these functions.
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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 for negative values of x, as it transforms the expression into a more manageable form for calculation.
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Rounding to a Specified Decimal Place
Rounding involves approximating a number to a certain number of decimal places, in this case, to the nearest thousandth (three decimal places). This skill is important to present answers clearly and consistently, especially when dealing with irrational or long decimal results from exponential calculations.
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Related Practice
Textbook Question
Determine whether each function graphed or defined is one-to-one. y = x+4 / x-3
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. (1/3)x = -3
Textbook Question
Determine whether each function graphed or defined is one-to-one. y = 2(x+1)2 - 6
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. See Example 1. g(2.34)
Textbook Question
Graph each function. ƒ(x) = 3x
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