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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 61
Chapter 5, Problem 61

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. 3 ln x + 5 ln y - 6 ln z

Verified step by step guidance
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Recall the logarithmic property that allows you to move coefficients in front of logarithms as exponents inside the logarithm: \(a \ln b = \ln b^a\).
Apply this property to each term: rewrite \(3 \ln x\) as \(\ln x^3\), \(5 \ln y\) as \(\ln y^5\), and \(-6 \ln z\) as \(\ln z^{-6}\).
Use the logarithmic property that the sum and difference of logarithms correspond to multiplication and division inside a single logarithm: \(\ln a + \ln b = \ln (a \cdot b)\) and \(\ln a - \ln b = \ln \left(\frac{a}{b}\right)\).
Combine the terms into a single logarithm: \(\ln \left( \frac{x^3 \cdot y^5}{z^6} \right)\).
Verify that the coefficient of the logarithm is 1, which it is, so the expression is now condensed into a single logarithm.

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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, quotient rule, and power rule. These allow combining or breaking 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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Change of Base Property

Condensing Logarithmic Expressions

Condensing logarithmic expressions means rewriting multiple logs into a single logarithm. This is done by applying the product, quotient, and power rules to combine terms, ensuring the final expression has a coefficient of 1.
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Expand & Condense Log Expressions

Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator involves recognizing values of logs based on known bases and properties, such as log base e (natural logs) of 1 is 0, or using simplifications from the condensed form to find exact values.
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Related Practice
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 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. (ln x)(ln 1) = 0

Textbook Question

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

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

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

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. 5 ln(2x)=20