Ch. 4 - Exponential and Logarithmic Functions

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Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

Problema 85

Capitolo 5, Problema 85

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 7=log 112

Guida verificata passo dopo passo

Start with the given equation: $2 \log x - \log 7 = \log 112$.

Use the logarithm power rule to rewrite $2 \log x$ as $\log x^2$, so the equation becomes $\log x^2 - \log 7 = \log 112$.

Apply the logarithm subtraction rule: $\log a - \log b = \log \left( \frac{a}{b} \right)$, to combine the left side into a single logarithm: $\log \left( \frac{x^2}{7} \right) = \log 112$.

Since $\log A = \log B$ implies $A = B$ (assuming the same base and valid domains), set the arguments equal: $\frac{x^2}{7} = 112$.

Solve for $x^2$ by multiplying both sides by 7, then take the square root of both sides to find $x$. Remember to check the domain restrictions for logarithms: $x$ must be positive.

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Properties of Logarithms

Understanding the properties of logarithms, such as the product, quotient, and power rules, is essential for simplifying and combining logarithmic expressions. For example, the difference of logs can be rewritten as the log of a quotient: log a - log b = log(a/b). These properties allow the equation to be manipulated into a solvable form.

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Change of Base Property

Domain of Logarithmic Functions

The domain of a logarithmic function includes only positive real numbers because the logarithm of zero or a negative number is undefined. When solving logarithmic equations, it is crucial to check that the solutions fall within the domain of the original expressions to avoid extraneous or invalid answers.

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Graphs of Logarithmic Functions

Solving Logarithmic Equations

Solving logarithmic equations often involves rewriting the equation using logarithm properties, isolating the logarithmic expression, and then converting the logarithmic form to its equivalent exponential form. This process helps find exact solutions, which can then be approximated using a calculator if needed.

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Solving Logarithmic Equations

Pratica correlata

Domanda del libro di testo

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)

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Domanda del libro di testo

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)

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Domanda del libro di testo

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?

649

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Domanda del libro di testo

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 8

894

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Domanda del libro di testo

Evaluate or simplify each expression without using a calculator. 10^{log 33}

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Domanda del libro di testo

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?

623

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