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 91

Capitolo 5, Problema 91

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(x−2)−ln(x+3)=ln(x−1)−ln(x+7)

Guida verificata passo dopo passo

Start with the given equation: \(\ln(x-2) - \ln(x+3) = \ln(x-1) - \ln(x+7)\).

Use the logarithmic property that \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\) to rewrite both sides: \(\ln \left( \frac{x-2}{x+3} \right) = \ln \left( \frac{x-1}{x+7} \right)\).

Since the natural logarithm function \(\ln(x)\) is one-to-one, set the arguments equal to each other: \(\frac{x-2}{x+3} = \frac{x-1}{x+7}\).

Cross-multiply to eliminate the fractions: \((x-2)(x+7) = (x-1)(x+3)\).

Expand both sides, simplify the resulting equation, and solve for \(x\). After finding the solutions, check each one to ensure it makes the arguments of all logarithms positive (i.e., \(x-2 > 0\), \(x+3 > 0\), \(x-1 > 0\), and \(x+7 > 0\)) to confirm they are in the domain.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Properties of Logarithms

Understanding the properties of logarithms, such as the difference rule ln(a) - ln(b) = ln(a/b), is essential for simplifying and solving logarithmic equations. These properties allow combining or breaking down logarithmic expressions to isolate the variable.

Video consigliato:

5:36

Change of Base Property

Domain of Logarithmic Functions

The domain of a logarithmic function includes only positive arguments. When solving equations like ln(x−2), the expressions inside the logarithms must be greater than zero, which restricts possible solutions and requires checking for extraneous roots.

Video consigliato:

5:26

Graphs of Logarithmic Functions

Solving Logarithmic Equations

Solving logarithmic equations often involves rewriting the equation using logarithm properties, exponentiating both sides to eliminate logs, and then solving the resulting algebraic equation. Verifying solutions against the domain is crucial to ensure validity.

Video consigliato:

5:02

Solving Logarithmic Equations

Pratica correlata

Domanda del libro di testo

Graph f(x) = 2^x and its inverse function in the same rectangular coordinate system.

1699

views

Domanda del libro di testo

n Exercises 92–93, rewrite the equation in terms of base e. Express the answer in terms of a natural logarithm and then round to three decimal places. y = 73(2.6)^x

664

views

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. ln(x−4)+ln(x+1)=ln(x−8)

1194

views

Domanda del libro di testo

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. log₄ (2x³) = 3 log₄ (2x)

787

views

Domanda del libro di testo

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. ln(8x³) = 3 ln (2x)

718

views

Domanda del libro di testo

Evaluate or simplify each expression without using a calculator. In (1/e⁶)

803

views