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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 21
Chapter 5, Problem 21

Evaluate each expression without using a calculator. log4 16

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1
Recognize that the expression \( \log_4 16 \) asks for the exponent to which the base 4 must be raised to get 16.
Express 16 as a power of 4. Since 16 is \( 4^2 \), rewrite the expression as \( \log_4 (4^2) \).
Use the logarithmic identity \( \log_b (b^k) = k \) to simplify \( \log_4 (4^2) \) to 2.
Conclude that \( \log_4 16 = 2 \) because 4 raised to the power 2 equals 16.
Thus, the value of the logarithm is the exponent 2.

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Key Concepts

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

Logarithm Definition

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log_b(a) = c means b^c = a. Understanding this definition is essential to evaluate logarithmic expressions.
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Properties of Exponents

Since logarithms are closely related to exponents, knowing how to express numbers as powers of a base helps simplify logarithms. For instance, recognizing that 16 = 4^2 allows you to rewrite log_4(16) as log_4(4^2).
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Logarithm Power Rule

The power rule states that log_b(a^n) = n * log_b(a). This property allows you to bring the exponent in the argument down as a multiplier, simplifying the evaluation of logarithms when the argument is a power of the base.
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Related Practice
Textbook Question

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}.

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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. logb (x2 y)

Textbook Question

In Exercises 19–29, evaluate each expression without using a calculator. If evaluation is not possible, state the reason. log16 4

Textbook Question

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}.

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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. log4(x64)\(\log\)_4\(\left\)(\(\frac{\sqrt{x}\)}{64}\(\right\))

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

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