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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 3
Chapter 5, Problem 3

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 5x=125

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1
Recognize that the equation is \(5^{x} = 125\). The goal is to express both sides as powers of the same base.
Recall that 125 can be written as a power of 5 because \(125 = 5^{3}\).
Rewrite the equation using this expression: \(5^{x} = 5^{3}\).
Since the bases are the same and the equation holds true, set the exponents equal to each other: \(x = 3\).
This gives the solution for \(x\) without needing to calculate any further.

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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 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.
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Expressing Numbers as Powers of the Same Base

To solve exponential equations, it is helpful to rewrite numbers as powers of a common base. For example, 125 can be expressed as 5³, which allows the equation 5^x = 125 to be rewritten as 5^x = 5^3.
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Equating Exponents

Once both sides of an exponential equation have the same base, the exponents can be set equal to each other because if a^m = a^n, then m = n. This property simplifies solving for the variable in the exponent.
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Rational Exponents
Related Practice
Textbook Question

In Exercises 1–10, approximate each number using a calculator. Round your answer to three decimal places. 3√5

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Textbook Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log7 (7x)

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Textbook Question

The graph of an exponential function is given. Select the function for each graph from the following options: f(x) = 4x, g(x) = 4-x, h(x) = -4-x, r(x) = -4-x+3

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Textbook Question

Write each equation in its equivalent exponential form. 2 = log9 x

Textbook Question

The graph of an exponential function is given. Select the function for each graph from the following options: f(x) = 4x, g(x) = 4-x, h(x) = -4-x, r(x) = -4-x+3

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Textbook Question

Write each equation in its equivalent exponential form. 2 = log3 x

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