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 19
Chapter 5, Problem 19

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)

Verified step by step guidance
1
Identify the bases on both sides of the equation: \(8^{(x+3)} = 16^{(x-1)}\). Notice that both 8 and 16 can be expressed as powers of 2.
Rewrite each base as a power of 2: \(8 = 2^3\) and \(16 = 2^4\). Substitute these into the equation to get \((2^3)^{(x+3)} = (2^4)^{(x-1)}\).
Apply the power of a power property: \((a^m)^n = a^{m \cdot n}\). This gives \(2^{3(x+3)} = 2^{4(x-1)}\).
Since the bases are the same (both are base 2), set the exponents equal to each other: \(3(x+3) = 4(x-1)\).
Solve the resulting linear equation for \(x\) by expanding both sides and isolating \(x\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

Key Concepts

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

Exponential Equations

An exponential equation is one in which variables appear as exponents. Solving these equations often involves rewriting expressions so that both sides have the same base, allowing the exponents to be set equal to each other.
Recommended video:
5:47
Solving Exponential Equations Using Logs

Expressing Numbers as Powers of the Same Base

To solve exponential equations, it is helpful to rewrite each number as a power of a common base. For example, 8 can be written as 2³ and 16 as 2⁴, enabling the comparison of exponents when bases match.
Recommended video:
05:10
Higher Powers of i

Equating Exponents

Once both sides of an equation have the same base, the exponents can be set equal to each other because if a^m = a^n, then m = n. This step transforms the problem into a simpler algebraic equation to solve for the variable.
Recommended video:
Guided course
04:06
Rational Exponents
Related Practice
Textbook Question

In Exercises 19–24, 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}.

Textbook Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. lnx5\(\ln\]\sqrt\)[5]{x}

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. e(x+1)=1/e

4
views
Textbook Question

In Exercises 16–18, write each equation in its equivalent logarithmic form. 13^y = 874

2
views
Textbook Question

Write each equation in its equivalent logarithmic form. 7y = 200

Textbook Question

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

3
views