Skip to main content
Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblem 21
Chapter 5, Problem 21

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. e(x+1)=1/e

Verified step by step guidance
1
Rewrite the right side of the equation \(e^{(x+1)} = \frac{1}{e}\) as a power of \(e\). Since \(\frac{1}{e} = e^{-1}\), the equation becomes \(e^{(x+1)} = e^{-1}\).
Now that both sides have the same base \(e\), set the exponents equal to each other: \(x + 1 = -1\).
Solve the resulting linear equation for \(x\) by isolating \(x\): subtract 1 from both sides to get \(x = -1 - 1\).
Simplify the right side to find the value of \(x\).
Verify your solution by substituting \(x\) back into the original equation to ensure both sides are equal.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

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

Exponential Functions and Their Properties

An exponential function has the form a^x, where the base a is a positive constant. Understanding how to manipulate and interpret these functions is essential for solving equations involving exponents, especially when the variable is in the exponent.
Recommended video:
6:13
Exponential Functions

Expressing Both Sides with the Same Base

To solve exponential equations, rewrite each side as powers of the same base. This allows you to set the exponents equal to each other, simplifying the equation to a linear or simpler form that can be solved algebraically.
Recommended video:
03:42
Linear Inequalities with Fractions & Variables on Both Sides

Properties of the Number e and Negative Exponents

The number e is a special irrational constant approximately equal to 2.718. Understanding that 1/e can be written as e^(-1) helps rewrite the equation with the same base, enabling the use of exponent rules to solve for the variable.
Recommended video:
4:47
The Number e
Related Practice
Textbook Question

The graph of an exponential function is given. Select the function for each graph from the following options:

f(x)=3x,g(x)=3x−1,h(x)=3x−1,f(x)=−3x,G(x)=3−x,H(x)=−3−x.f(x) = 3^x, \(\quad\) g(x) = 3^{x-1}, \(\quad\) h(x) = 3^x - 1, \(\f\)(x) = -3^x, \(\quad\) G(x) = 3^{-x}, \(\quad\) H(x) = -3^{-x}.

1025
views
Textbook Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. logb (x2 y)

821
views
Textbook Question

In Exercises 19–29, evaluate each expression without using a calculator. If evaluation is not possible, state the reason. log16 4

904
views
Textbook Question

Evaluate each expression without using a calculator. log4 16

943
views
Textbook Question

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 8(x+3)=16(x−1)

970
views
Textbook Question

In Exercises 19–29, evaluate each expression without using a calculator. If evaluation is not possible, state the reason. log5 (1/5)

917
views