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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 25
Chapter 5, Problem 25

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

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1
Recognize that the equation is \(\left(\frac{1}{3}\right)^x = -3\). Here, the base \(\frac{1}{3}\) is a positive number less than 1, and the expression represents an exponential function.
Recall that an exponential function with a positive base (other than 1) raised to any real power will always yield a positive result. This means \(\left(\frac{1}{3}\right)^x > 0\) for all real values of \(x\).
Since the right side of the equation is \(-3\), which is negative, there is no real value of \(x\) that can satisfy the equation \(\left(\frac{1}{3}\right)^x = -3\).
Conclude that the equation has no real solutions because an exponential function with a positive base cannot equal a negative number.
If complex solutions are considered, you would need to use logarithms with complex numbers, but since this is a College Algebra problem focusing on real solutions, the answer is that no real solution exists.

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Key Concepts

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

Properties of Exponential Functions

Exponential functions have the form a^x where the base a is positive and not equal to 1. Their outputs are always positive numbers, meaning expressions like (1/3)^x cannot equal a negative number. Understanding this helps identify when an equation has no real solution.
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Domain and Range of Exponential Functions

The domain of an exponential function is all real numbers, but its range is strictly positive real numbers (0, ∞). This means exponential expressions cannot produce zero or negative values, which is crucial when solving equations involving exponentials.
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Domain & Range of Transformed Functions

Solving Exponential Equations

To solve exponential equations, isolate the exponential expression and apply logarithms if necessary. However, if the equation sets an exponential equal to a negative number, no real solution exists because exponentials never output negatives.
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Solving Exponential Equations Using Logs
Related Practice
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Find each value. If applicable, give an approximation to four decimal places. log 518 - log 342

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Determine whether each function graphed or defined is one-to-one. y = x+4 / x-3

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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. g(-1.68)

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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(2.34)

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Graph each function. ƒ(x) = 3x

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