Ch. 4 - Exponential and Logarithmic Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

Problem 51

Chapter 5, Problem 51

In Exercises 50–53, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $\log_{4} (\frac{\sqrt{x}}{64})$

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Recall the logarithmic expression given: $\log_4 \left( \frac{\sqrt{x}}{64} \right)$.

Use the logarithm property for division: $\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N$. So rewrite the expression as $\log_4 (\sqrt{x}) - \log_4 (64)$.

Express the square root as an exponent: $\sqrt{x} = x^{\frac{1}{2}}$. Then apply the power rule for logarithms: $\log_b (M^p) = p \log_b M$. So $\log_4 (\sqrt{x}) = \frac{1}{2} \log_4 x$.

Rewrite 64 as a power of 4 if possible. Since $4^3 = 64$, we have $\log_4 (64) = \log_4 (4^3)$.

Apply the power rule again: $\log_4 (4^3) = 3 \log_4 4$. Since $\log_4 4 = 1$, this simplifies to 3. So the expression becomes $\frac{1}{2} \log_4 x - 3$.

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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^p) = p * log_b(M). These properties are essential for simplifying and expanding logarithmic expressions.

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 helps apply logarithmic power rules effectively, enabling the expansion of logarithmic expressions involving roots.

Evaluating Logarithms with Known Bases

Evaluating logarithms without a calculator often involves recognizing numbers as powers of the base. For example, 64 is 4 raised to the 3rd power (4^3). This allows simplification of expressions like log_4(64) by rewriting 64 in terms of the base 4, making evaluation straightforward.

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Textbook Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. log x + 3 log y

Textbook Question

Graph functions f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. f(x) = 3^x and g(x) = (1/3). 3^x

Textbook Question

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. ln x=2

Textbook Question

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Textbook Question

In Exercises 50–53, 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}\(\right\))

Verified video answer for a similar problem:

Key Concepts

Properties of Logarithms

Radicals and Exponents

Evaluating Logarithms with Known Bases

In Exercises 50–53, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $\log_{4} (\frac{\sqrt{x}}{64})$