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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 53
Chapter 5, Problem 53

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. 2 logb x + 3 logb y

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Identify the given expression: \(2 \log_{b} x + 3 \log_{b} y\).
Recall the logarithmic property that allows you to move coefficients as exponents inside the logarithm: \(a \log_{b} M = \log_{b} (M^{a})\).
Apply this property to each term: \(2 \log_{b} x = \log_{b} (x^{2})\) and \(3 \log_{b} y = \log_{b} (y^{3})\).
Use the logarithmic property for addition: \(\log_{b} A + \log_{b} B = \log_{b} (A \times B)\) to combine the two terms into a single logarithm.
Write the final condensed expression as \(\log_{b} (x^{2} y^{3})\), which is a single logarithm with coefficient 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 rules such as the product, quotient, and power rules. These allow combining multiple logarithmic terms into a single logarithm by converting coefficients into exponents and combining sums or differences into products or quotients.
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Power Rule of Logarithms

The power rule states that a coefficient in front of a logarithm can be rewritten as an exponent inside the logarithm, i.e., a·log_b(x) = log_b(x^a). This is essential for condensing expressions with coefficients into a single logarithm.
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Power Rules

Condensing Logarithmic Expressions

Condensing logarithmic expressions means rewriting a sum or difference of logarithms as a single logarithm. This involves applying the product or quotient rules after using the power rule to handle coefficients, simplifying the expression into one logarithm with coefficient 1.
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Related Practice
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. g(x) = log₂ (x + 1)

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) = (½)x and g(x) = (½)x-1 + 1

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 a. compounded semiannually

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. log4(x+5)=3

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. lnxe3\(\ln\]\sqrt\)[3]{\(\frac{x}{e}\)}

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. (1/2)ln x + ln y