Question

Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. $$3^{x}=-4$$

Name: Problem 20, Chapter 5: A Graphical Approach to College Algebra
Uploaded: 2020-07-18T07:00:30-08:00
Duration: 19 s
Channel: Ankit Gupta
Description: Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. 3^x=-4

   Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator.
$$3^{x}=-4$$

A Graphical Approach to College Algebra

John… 6th Edition

Chapter 5, Problem 20

Instant Answer

Solved by Expert Ankit Gupta

07/18/2020

Step 1

An exponential function is a function of the form $f(x) = a^{x}$, where $a$ is a positive real number. The graph of an exponential function always lies above the x-axis, meaning that the output of an exponential function is always positive. Show more…

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Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. $$3^{x}=-4$$

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Key Concepts

Complex Numbers in Exponential Equations

When an exponential equation equates to a negative number, one may consider solutions in the complex number system. Using the concept of complex logarithms, one can express the solution in the form involving both real and imaginary parts. This approach extends the idea of logarithms to the complex plane and introduces periodicity, resulting in an infinite set of complex solutions.

Domain of Exponential Functions

Exponential functions with a positive base are defined for all real inputs, and their outputs are always positive. This means that for any such exponential function, it is impossible to obtain a negative number when the exponent is a real number. Recognizing this property is fundamental when solving exponential equations, as it immediately tells us if a proposed equation has a real solution or not.

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