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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 63
Chapter 5, Problem 63

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)(log x + log y)

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Start with the given expression: \(\frac{1}{2}(\log x + \log y)\).
Use the logarithm property that the sum of logarithms is the logarithm of the product: \(\log x + \log y = \log(xy)\).
Rewrite the expression as \(\frac{1}{2} \log(xy)\).
Apply the power rule of logarithms, which states that \(a \log b = \log(b^a)\), to move the coefficient inside the logarithm: \(\frac{1}{2} \log(xy) = \log((xy)^{\frac{1}{2}})\).
Express the final condensed form as a single logarithm: \(\log \sqrt{xy}\).

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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 rule (log a + log b = log(ab)), the power rule (k log a = log(a^k)), and the quotient rule (log a - log b = log(a/b)). These rules allow combining or breaking down logarithmic expressions to simplify or condense them.
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Condensing Logarithmic Expressions

Condensing logarithmic expressions means rewriting multiple logarithms as a single logarithm. This involves applying the properties of logarithms to combine sums, differences, and coefficients into one logarithmic term with coefficient 1, making the expression simpler and easier to evaluate.
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Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator requires recognizing logarithms of numbers that are powers of the base, such as log base 10 of 100 = 2. Simplifying expressions using properties can help rewrite logarithms in terms of known values, enabling exact evaluation without computational tools.
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Related Practice
Textbook Question

Give the equation of each exponential function whose graph is shown.

Textbook Question

In Exercises 60–63, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. (log2 x)^4 = 4 log2 x

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Textbook Question

In Exercises 64–73, solve each exponential equation. Where necessary, express the solution set in terms of natural or common logarithms and use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 2^(4x-2) = 64

Textbook Question

The figure shows the graph of f(x) = log x. In Exercises 59–64, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. h(x) = log x − 1

Textbook Question

The figure shows the graph of f(x) = log x. In Exercises 59–64, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range.

g(x) = 1-log x

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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. 6+2 ln x=5