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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 96a
Chapter 5, Problem 96a

Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given ƒ(x) = 3x, find ƒ(log3 2)

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Recognize that the function is given as \(f(x) = 3^x\), and you need to find \(f(\log_3 2)\), which means substituting \(x\) with \(\log_3 2\) in the function.
Write the expression explicitly as \(f(\log_3 2) = 3^{\log_3 2}\).
Recall the property of exponents and logarithms: for any positive base \(a \neq 1\), \(a^{\log_a b} = b\). This is because the logarithm \(\log_a b\) is the exponent to which \(a\) must be raised to get \(b\).
Apply this property to simplify \(3^{\log_3 2}\) directly to \(2\) without further calculation.
Conclude that \(f(\log_3 2) = 2\) based on the exponential and logarithmic inverse relationship.

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Key Concepts

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

Exponential Functions

An exponential function has the form f(x) = a^x, where the base a is a positive constant not equal to 1. It models growth or decay processes and has unique properties, such as the function being one-to-one and always positive. Understanding how to evaluate and manipulate these functions is essential for solving problems involving exponents.
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Exponential Functions

Logarithmic Functions

A logarithmic function is the inverse of an exponential function and is written as log_a(x), where a is the base. It answers the question: to what power must the base a be raised to get x? Knowing how to interpret and use logarithms is crucial for simplifying expressions and solving equations involving exponents.
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Inverse Function Property of Exponentials and Logarithms

Exponential and logarithmic functions with the same base are inverses, meaning f(log_a(x)) = x and log_a(a^x) = x. This property allows simplification of expressions like f(log_3 2) by directly substituting and canceling the functions, which is key to evaluating the given expression efficiently.
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Related Practice
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