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 65

Chapter 5, Problem 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. $\ln \sqrt{x + 3} = 1$

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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Related Practice

Textbook Question

The figure shows the graph of f(x) = ln x. In Exercises 65–74, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. g(x) = ln (x+2)

Textbook Question

In Exercises 64–73, solve each exponential equation. Where necessary, express the solution set in terms of natural or common logarithms and use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 2^(4x-2) = 64

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. $$\frac{1}{2}$ $\left$( $\log$_5 x + $\log$_5 y $\right$) - 2 $\log$_5 (x + 1)$

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. $$\frac{1}{3}$ $\left$[ 2 $\ln$(x + 5) - $\ln$ x - $\ln$ (x^2 - 4) $\right$]$

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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. log₅ x+log₅(4x−1)=1

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Textbook Question

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