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 39

Chapter 5, Problem 39

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log$ $\left$( $\frac{10x^2 \sqrt[3]{1 - x}$}{7(x + 1)^2} $\right$)$

Verified step by step guidance

Start by recognizing that the logarithm of a quotient can be expressed as the difference of logarithms: $\log \left( \frac{A}{B} \right) = \log A - \log B$.

Apply this property to the given expression: $\log \left( \frac{10x^{2} \sqrt[3]{1 - x}}{7(x + 1)^{2}} \right) = \log \left( 10x^{2} \sqrt[3]{1 - x} \right) - \log \left( 7(x + 1)^{2} \right)$.

Next, use the product property of logarithms: $\log (AB) = \log A + \log B$, to expand both logarithms: $\log 10 + \log x^{2} + \log \sqrt[3]{1 - x} - \left( \log 7 + \log (x + 1)^{2} \right)$.

Rewrite the logarithms of powers using the power property: $\log (a^{b}) = b \log a$. So, $\log x^{2} = 2 \log x$, $\log \sqrt[3]{1 - x} = \log (1 - x)^{1/3} = \frac{1}{3} \log (1 - x)$, and $\log (x + 1)^{2} = 2 \log (x + 1)$.

Combine all parts to write the fully expanded expression: $\log 10 + 2 \log x + \frac{1}{3} \log (1 - x) - \log 7 - 2 \log (x + 1)$.

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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 the product, quotient, and power rules, which allow the expansion or simplification of logarithmic expressions. For example, log(ab) = log a + log b, log(a/b) = log a - log b, and log(a^n) = n log a. These rules help break down complex expressions into simpler sums and differences of logs.

Radicals and Exponents

Radicals such as cube roots can be expressed as fractional exponents, e.g., ∛(1 - x) = (1 - x)^(1/3). Understanding how to rewrite radicals as exponents allows the use of logarithm power rules to simplify expressions involving roots and powers.

Evaluating Logarithms Without a Calculator

Some logarithmic values can be simplified or evaluated exactly using known log values and properties, especially when the arguments are products or powers of numbers like 10. Recognizing these can help simplify expressions without a calculator, such as log(10) = 1.

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Related Practice

Textbook Question

In Exercises 39–40, graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. f(x) = log x and g(x) = - log (x+3)

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 5 + log 2

Textbook Question

The figure shows the graph of f(x) = e^x. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn h(x) = e^x-1+2

Textbook Question

Evaluate each expression without using a calculator. log₅ 5⁷

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. 7^0.3x=813

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log(10x21−x37(x+1)2)\(\log\) \(\left\)( \(\frac{10x^2 \sqrt[3]{1 - x}\)}{7(x + 1)^2} \(\right\))log(7(x+1)210x231−x)

Verified video answer for a similar problem:

Key Concepts

Properties of Logarithms

Radicals and Exponents

Evaluating Logarithms Without a Calculator

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $\(\log\) \(\left\)( \(\frac{10x^2 \sqrt[3]{1 - x}\)}{7(x + 1)^2} \(\right\))$