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 57

Chapter 5, Problem 57

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

Verified step by step guidance

Identify the logarithmic expression given: \(\frac{1}{2} \ln x - \ln y\).

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

Apply this property to the first term: \(\frac{1}{2} \ln x = \ln x^{\frac{1}{2}}\).

Use the logarithmic property for subtraction: \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\).

Combine the terms into a single logarithm: \(\ln \left( \frac{x^{\frac{1}{2}}}{y} \right)\).

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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 us to combine or break 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.

Power Rule of Logarithms

The power rule states that a coefficient multiplied by a logarithm can be expressed as an exponent inside the logarithm: a * log_b(x) = log_b(x^a). This is essential for rewriting expressions like (1/2) ln x as ln(x^(1/2)) to help condense the expression.

Combining Logarithms Using Quotient Rule

The quotient rule states that the difference of two logarithms with the same base can be written as the logarithm of a quotient: log_b(A) - log_b(B) = log_b(A/B). This allows us to combine terms like ln(x^(1/2)) - ln(y) into a single logarithm ln(x^(1/2)/y).

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Product, Quotient, and Power Rules of Logs

Related Practice

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. 3 ln x - (1/3) ln y

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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

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

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