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 55

Chapter 5, Problem 55

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

Verified step by step guidance

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 $5 \ln x$ as $\ln x^5$ and $-2 \ln y$ as $\ln y^{-2}$.

Use the logarithmic property that the difference of logarithms is the logarithm of a quotient: $\ln a - \ln b = \ln \left( \frac{a}{b} \right)$.

Combine the two logarithms into a single logarithm: $\ln x^5 - \ln y^2 = \ln \left( \frac{x^5}{y^2} \right)$.

Write the final condensed expression as a single logarithm with coefficient 1: $\ln \left( \frac{x^5}{y^2} \right)$.

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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 combining or breaking down logarithmic expressions. For example, the power rule states that a coefficient in front of a logarithm can be rewritten as an exponent inside the log, i.e., a·ln(b) = ln(b^a).

Condensing Logarithmic Expressions

Condensing logarithmic expressions means rewriting multiple logarithms as a single logarithm. This is done by applying the product rule (log a + log b = log(ab)), quotient rule (log a - log b = log(a/b)), and power rule to combine terms into one logarithm with coefficient 1.

Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator involves recognizing values that simplify to known logarithmic results, such as ln(e) = 1 or ln(1) = 0. Simplifying expressions using properties can sometimes reduce the argument to these known values, allowing exact evaluation.

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Related Practice

Textbook Question

In Exercises 54–57, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is $1$ . $log 3 minus 3 log x$

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Textbook Question

In Exercises 53-58, 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) = (1/2)log₂ x

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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) = 2 + log₂x

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Textbook Question

Graph f and g in the same rectangular coordinate system. Then find the point of intersection of the two graphs. f(x) = 2^x, g(x) = 2^-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. log₂(x+25)=4