Ch. 4 - Inverse, Exponential, and Logarithmic Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

Problem 96c

Chapter 5, Problem 96c

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

Verified step by step guidance

Recognize that the function is given as \(f(x) = 3^x\) and you need to evaluate \(f(\log_3(2 \ln 3))\).

Recall the definition of the function evaluation: \(f(\log_3(2 \ln 3)) = 3^{\log_3(2 \ln 3)}\).

Use the property of exponents and logarithms that states \(a^{\log_a(b)} = b\) for any positive \(a \neq 1\) and \(b > 0\).

Apply this property to simplify \(3^{\log_3(2 \ln 3)}\) directly to \(2 \ln 3\).

Thus, the expression \(f(\log_3(2 \ln 3))\) simplifies to \(2 \ln 3\) without further calculation.

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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 properties such as a^(m+n) = a^m * a^n. Understanding how to manipulate and evaluate these functions is essential for solving problems involving exponents.

Logarithmic Functions

A logarithmic function is the inverse of an exponential function, defined as log_a(x), which answers the question: to what power must the base a be raised to get x? Key properties include log_a(a^x) = x and a^(log_a(x)) = x. Recognizing these inverse relationships helps simplify expressions involving logs and exponents.

Properties of Logarithms and Exponents

The properties of logarithms and exponents, such as the inverse relationship a^(log_a(x)) = x and the ability to rewrite expressions using these properties, are crucial for simplifying complex expressions. For example, evaluating f(log_3(2 ln 3)) involves applying these properties to simplify the composition of functions.

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

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Solve each equation for the indicated variable. Use logarithms with the appropriate bases. A = P (1 + r/n)^tn, for t

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Textbook Question

Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given g(x) = e^x, find g(ln 1/e)

Textbook Question

Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). Given ƒ(x) = log₂ x, find ƒ(2⁷)

Textbook Question

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

Textbook Question

Given that log₁₀ 2 ≈ 0.3010 and log₁₀ 3 ≈ 0.4771, find each logarithm without using a calculator. log₁₀ 9/4

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 ln 3))

Verified video answer for a similar problem:

Key Concepts

Exponential Functions

Logarithmic Functions

Properties of Logarithms and Exponents

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