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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 24
Chapter 5, Problem 24

Evaluate each expression without using a calculator. log3 27

Verified step by step guidance
1
Recognize that the expression is \( \log_{3} 27 \), which asks: "To what power must 3 be raised to get 27?"
Rewrite 27 as a power of 3. Since \( 27 = 3^3 \), substitute this into the logarithm: \( \log_{3} 3^3 \).
Use the logarithmic identity \( \log_{a} a^x = x \) to simplify the expression.
Apply the identity to get \( \log_{3} 3^3 = 3 \).
Conclude that the value of \( \log_{3} 27 \) is the exponent 3, since \( 3^3 = 27 \).

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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 base 3 of 27 asks, '3 raised to what power equals 27?' Understanding this definition is essential to evaluate logarithmic expressions.
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Exponentiation and Powers

Recognizing powers of numbers helps simplify logarithms. Since 27 is 3 raised to the 3rd power (3^3 = 27), this relationship allows direct evaluation of log base 3 of 27 by identifying the exponent.
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Properties of Logarithms

Logarithms have properties such as log_b(b^x) = x, which means the logarithm of a base raised to an exponent is just that exponent. This property simplifies evaluating expressions like log_3(27) by converting the problem into finding the exponent.
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Related Practice
Textbook Question

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 10x=3.91

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Textbook Question

Evaluate each expression without using a calculator. log2 64

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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)=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

Evaluate each expression without using a calculator. log5 (1/5)

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Textbook Question

In Exercises 19–29, evaluate each expression without using a calculator. If evaluation is not possible, state the reason. ln e5

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Textbook Question

Begin by graphing f(x) = 2x. Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. g(x) = 2x+1

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