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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 23
Chapter 5, Problem 23

Evaluate each expression without using a calculator. log2 64

Verified step by step guidance
1
Recognize that the expression \( \log_2 64 \) asks for the exponent to which the base 2 must be raised to get 64.
Express 64 as a power of 2. Since \( 64 = 2^6 \), rewrite the expression as \( \log_2 (2^6) \).
Use the logarithmic identity \( \log_b (b^x) = x \) to simplify \( \log_2 (2^6) \) to 6.
Therefore, \( \log_2 64 = 6 \), meaning 2 raised to the 6th power equals 64.
This completes the evaluation without using a calculator.

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Key Concepts

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

Logarithms and Their Definition

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log base 2 of 64 asks, '2 raised to what power equals 64?' Understanding this definition is essential to evaluate logarithmic expressions.
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Properties of Exponents

Since logarithms are the inverse of exponents, knowing how to express numbers as powers of a base helps in evaluating logs. For instance, recognizing that 64 = 2^6 allows you to rewrite the logarithm in terms of exponents and solve it easily.
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Evaluating Logarithms Without a Calculator

To evaluate logarithms without a calculator, rewrite the argument as a power of the base and then use the definition of logarithms. This method relies on mental math and familiarity with common powers, enabling quick and accurate solutions.
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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. log3 27

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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. log16 4

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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

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}\(\right\))

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Textbook Question

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

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