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 83

Chapter 5, Problem 83

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. log(x+4)−log 2=log(5x+1)

Verified step by step guidance

Start with the given equation: $\log(x+4) - \log 2 = \log(5x+1)$.

Use the logarithmic property that states $\log a - \log b = \log \left( \frac{a}{b} \right)$ to combine the left side: $\log \left( \frac{x+4}{2} \right) = \log(5x+1)$.

Since $\log A = \log B$ implies $A = B$ (assuming the bases are the same and the arguments are positive), set the arguments equal: $\frac{x+4}{2} = 5x + 1$.

Solve the resulting equation for $x$ by multiplying both sides by 2 to clear the denominator: $x + 4 = 2(5x + 1)$, then simplify and isolate $x$.

Check the domain restrictions: ensure that the arguments of all logarithms are positive, so $x + 4 > 0$ and $5x + 1 > 0$. Reject any solution that does not satisfy these conditions.

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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, the quotient rule allows combining or separating logarithmic expressions, e.g., log(a) - log(b) = log(a/b), which simplifies the equation for easier solving.

Domain of Logarithmic Functions

The domain of a logarithmic function includes only positive arguments because the logarithm of zero or a negative number is undefined. When solving logarithmic equations, it is crucial to check that solutions keep all log arguments positive to ensure valid answers.

Solving Logarithmic Equations

Solving logarithmic equations often involves rewriting the equation using logarithm properties, then converting to an exponential form to isolate the variable. After finding potential solutions, verify each against the domain restrictions to reject extraneous roots.

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Related Practice

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. 2 log x=log 25

Textbook Question

In Exercises 83–88, let log_b 2 = A and log_b 3 = C and Write each expression in terms of A and C.

log_b (3/2)

Textbook Question

Use the formula for continuous compounding to solve Exercises 84–85. How long, to the nearest tenth of a year, will it take \$50,000 to triple in value at an annual rate of 7.5% compounded continuously?

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Textbook Question

Expand: $l o g sub 8 of open paren 4 square root x over 64 y 3 close paren$

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Textbook Question

Evaluate or simplify each expression without using a calculator. log 10⁷

Textbook Question

Use the formula for continuous compounding to solve Exercises 84–85. What annual rate, to the nearest percent, is required for an investment subject to continuous compounding to triple in 5 years?

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