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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblema 57
Capitolo 5, Problema 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

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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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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.
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Change of Base Property

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.
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Power Rules

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

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In Exercises 58–59, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log4 0.863

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