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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 15
Chapter 5, Problem 15

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. 0.8x = 4

Verified step by step guidance
1
Identify the equation to solve: \(0.8^x = 4\).
Since the variable \(x\) is in the exponent, apply the natural logarithm (or log base 10) to both sides to bring the exponent down: \(\ln(0.8^x) = \ln(4)\).
Use the logarithm power rule to rewrite the left side: \(x \cdot \ln(0.8) = \ln(4)\).
Isolate \(x\) by dividing both sides by \(\ln(0.8)\): \(x = \frac{\ln(4)}{\ln(0.8)}\).
Calculate the right-hand side using a calculator to find the decimal approximation of \(x\) to the nearest thousandth.

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Key Concepts

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

Exponential Equations

An exponential equation involves variables in the exponent, such as 0.8^x = 4. Solving these requires understanding how to manipulate expressions where the unknown is an exponent, often by applying logarithms to isolate the variable.
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Logarithms and Their Properties

Logarithms are the inverse operations of exponentials and are used to solve equations where the variable is an exponent. Applying logarithms to both sides of an equation allows you to bring the exponent down and solve for the variable.
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Change of Base Property

Decimal Approximation of Irrational Numbers

When solutions involve irrational numbers, they cannot be expressed exactly as fractions or decimals. Instead, they are approximated to a certain decimal place, such as rounding to the nearest thousandth, to provide a practical numerical answer.
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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. (1/2)x = 5

Textbook Question

Find each value. If applicable, give an approximation to four decimal places. log 63

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(3)

Textbook Question

Find each value. If applicable, give an approximation to four decimal places. log 0.1

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. g(2)

Textbook Question

Determine whether each function graphed or defined is one-to-one.

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