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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

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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Recall the logarithmic property for quotients: \(\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N\). Apply this to rewrite \(\log_4 \left( \frac{\sqrt{x}}{64} \right)\) as \(\log_4 (\sqrt{x}) - \log_4 (64)\).
Express the square root as an exponent: \(\sqrt{x} = x^{\frac{1}{2}}\). Use the power rule of logarithms: \(\log_b (M^p) = p \log_b M\). So, \(\log_4 (\sqrt{x}) = \log_4 (x^{\frac{1}{2}}) = \frac{1}{2} \log_4 x\).
Rewrite 64 as a power of 4 if possible. Since \(4^3 = 64\), we have \(64 = 4^3\). Use the power rule again: \(\log_4 (64) = \log_4 (4^3) = 3 \log_4 4\).
Recall that \(\log_b b = 1\) for any base \(b\). Therefore, \(\log_4 4 = 1\), so \(\log_4 (64) = 3 \times 1 = 3\).
Combine all parts to write the expanded form: \(\log_4 \left( \frac{\sqrt{x}}{64} \right) = \frac{1}{2} \log_4 x - 3\).

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Key Concepts

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

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow us to rewrite logarithmic expressions by expanding or condensing them. For example, log_b(M/N) = log_b(M) - log_b(N) and log_b(M^k) = k * log_b(M). These properties are essential for simplifying and expanding logarithmic expressions.
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Change of Base Property

Radicals and Exponents

Understanding how to express radicals as fractional exponents is crucial. For instance, the square root of x can be written as x^(1/2). This conversion allows the use of logarithm power rules to simplify expressions involving roots. Recognizing this helps in expanding logarithmic expressions involving radicals.
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Change of Base and Evaluating Logarithms

Evaluating logarithms without a calculator often involves expressing numbers as powers of the base. For example, 64 can be written as 4^3 since 4^3 = 64. This allows direct simplification of logarithmic terms like log_4(64) = 3. This concept helps in simplifying and evaluating logarithmic expressions exactly.
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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

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

Textbook Question

Evaluate each expression without using a calculator. log2 64

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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

Textbook Question

Evaluate each expression without using a calculator. log4 16

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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