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 - (1/3) ln y

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Recall the logarithmic property that allows you to move coefficients in front of logarithms as exponents inside the log: \(a \ln b = \ln b^{a}\).

Apply this property to each term: rewrite \(3 \ln x\) as \(\ln x^{3}\) and \(\frac{1}{3} \ln y\) as \(\ln y^{\frac{1}{3}}\).

Rewrite the original expression using these new forms: \(\ln x^{3} - \ln y^{\frac{1}{3}}\).

Use the logarithmic subtraction property: \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\) to combine the two logarithms into one.

Write the final condensed expression as a single logarithm: \(\ln \left( \frac{x^{3}}{y^{\frac{1}{3}}} \right)\).

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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 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, which is essential for condensing expressions.

Condensing Logarithmic Expressions

Condensing logarithmic expressions means rewriting multiple logarithms as a single logarithm. This involves applying the properties of logarithms to combine terms, such as turning sums into products and differences into quotients inside the log. The goal is to have one logarithm with a coefficient of 1.

Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator involves recognizing logarithmic values based on known log properties or special values, such as log base 10 of 1 is 0, or log base e of e is 1. Simplifying expressions using these known values helps in exact evaluation when possible.

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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. log₃(x+4)=−3

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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

Textbook Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. (1/2)ln x - ln y

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Textbook Question

Graph y= 2^x and x = 2^y in the same rectangular coordinate system.

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

Textbook Question