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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 55
Chapter 5, Problem 55

In Exercises 54–57, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 11. log3−3logx\(\log\)3-3\(\log\) x

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Identify the given expression: \(\log 3 - 3 \log x\).
Recall the logarithmic property that allows you to move coefficients as exponents: \(a \log b = \log b^{a}\).
Apply this property to the term \(-3 \log x\), rewriting it as \(\log x^{-3}\).
Rewrite the expression using the property: \(\log 3 - \log x^{3}\).
Use the logarithmic subtraction property: \(\log a - \log b = \log \left( \frac{a}{b} \right)\) to combine into a single logarithm: \(\log \left( \frac{3}{x^{3}} \right)\).

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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 combine or break down logarithmic expressions. For example, the power rule states that a coefficient in front of a log can be rewritten as an exponent inside the log.
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Power Rule of Logarithms

The power rule states that a coefficient multiplied by a logarithm can be expressed as the logarithm of the argument raised to that coefficient. For instance, a·log_b(x) = log_b(x^a). This is essential for rewriting expressions to have a single logarithm.
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Combining Logarithmic Expressions

To condense multiple logarithmic terms into one, use the product rule (log_b(M) + log_b(N) = log_b(M·N)) and quotient rule (log_b(M) - log_b(N) = log_b(M/N)). Applying these rules helps write the expression as a single logarithm with coefficient 1.
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Related Practice
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. 5 ln x - 2 ln y

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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. log2(x+25)=4

Textbook Question

Use the compound interest formulas A = P (1+ r/n)nt and A =Pert to solve exercises 53-56. Round answers to the nearest cent. Find the accumulated value of an investment of \$10,000 for 5 years at an interest rate of 1.32% if the money is d. compounded continuously.

Textbook Question

Use the compound interest formulas A = P (1+ r/n)nt and A =Pert to solve exercises 53-56. Round answers to the nearest cent. Find the accumulated value of an investment of \$10,000 for 5 years at an interest rate of 1.32% if the money is c. compounded monthly.

Textbook Question

Use the compound interest formulas A = P (1+ r/n)nt and A =Pert to solve exercises 53-56. Round answers to the nearest cent. Suppose that you have \$12,000 to invest. Which investment yields the greater return over 3 years: 0.96% compounded monthly or 0.95% compounded continuously?

Textbook Question

Begin by graphing f(x) = log₂ x. Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. h(x)=1+ log₂ x