Ch. 4 - Exponential and Logarithmic Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo

Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

Problema 65

Capitolo 5, Problema 65

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. $\ln \sqrt{x + 3} = 1$

Guida verificata passo dopo passo

Rewrite the equation $ \ln \sqrt{x} + 3 = 1 $ by isolating the logarithmic term: subtract 3 from both sides to get $ \ln \sqrt{x} = 1 - 3 $.

Simplify the right side: $ 1 - 3 = -2 $, so the equation becomes $ \ln \sqrt{x} = -2 $.

Recall that $ \sqrt{x} = x^{\frac{1}{2}} $, so rewrite the logarithm as $ \ln x^{\frac{1}{2}} = -2 $.

Use the logarithmic power rule: $ \ln x^{\frac{1}{2}} = \frac{1}{2} \ln x $, so the equation becomes $ \frac{1}{2} \ln x = -2 $.

Multiply both sides by 2 to isolate $ \ln x $: $ \ln x = -4 $. Then, rewrite in exponential form: $ x = e^{-4} $. Finally, check the domain by ensuring $ x > 0 $ since the logarithm is defined only for positive arguments.

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Properties of Logarithms

Understanding the properties of logarithms, such as the product, quotient, and power rules, is essential. In this problem, recognizing that ln(√x) can be rewritten as (1/2)ln(x) helps simplify the equation and solve for x.

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Change of Base Property

Domain of Logarithmic Functions

The domain of a logarithmic function includes only positive real numbers inside the log. When solving, it is crucial to check that the solution values keep the argument of the logarithm positive to ensure the solution is valid.

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Graphs of Logarithmic Functions

Solving Exponential Equations

After isolating the logarithmic expression, converting the equation from logarithmic to exponential form allows solving for x. For example, if ln(y) = k, then y = e^k, which helps find the exact solution.

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Solving Exponential Equations Using Logs

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