Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

6. Exponential & Logarithmic Functions

Properties of Logarithms

College Algebra

6. Exponential & Logarithmic Functions

Properties of Logarithms

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1:27 minutes

Problem 96

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(log_3 2)

Verified step by step guidance

Identify the function given: \( f(x) = 3^x \).

Recognize that you need to evaluate \( f(\log_3 2) \).

Recall the property of logarithms: \( \log_b a = c \) implies \( b^c = a \).

Apply the property: since \( \log_3 2 \) is the exponent to which 3 must be raised to get 2, it follows that \( 3^{\log_3 2} = 2 \).

Conclude that \( f(\log_3 2) = 2 \) by substituting back into the function.

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Key Concepts

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

Exponential Functions

Exponential functions are mathematical expressions in the form ƒ(x) = a^x, where 'a' is a positive constant and 'x' is the variable exponent. These functions exhibit rapid growth or decay, depending on the base 'a'. Understanding their properties, such as the behavior of the function as 'x' approaches positive or negative infinity, is crucial for evaluating expressions involving exponents.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions and are expressed as log_b(a) = c, meaning b^c = a. They help in solving equations where the variable is an exponent. The properties of logarithms, such as the product, quotient, and power rules, are essential for manipulating and simplifying logarithmic expressions, particularly when evaluating logarithms of different bases.

Change of Base Formula

The change of base formula allows us to convert logarithms from one base to another, expressed as log_b(a) = log_k(a) / log_k(b) for any positive base 'k'. This is particularly useful when dealing with logarithms of bases that are not easily computable. In the context of the given problem, applying this formula can simplify the evaluation of ƒ(log_3 2) by converting it into a more manageable form.