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 75

Capitolo 5, Problema 75

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. log₂(x−6)+log₂(x−4)−log₂ x=2

Guida verificata passo dopo passo

Identify the domain restrictions for the logarithmic expressions: since the arguments of the logarithms must be positive, set up the inequalities \(x - 6 > 0\), \(x - 4 > 0\), and \(x > 0\). Solve these to find the domain of \(x\).

Use the logarithmic property that \(\log_b A + \log_b B = \log_b (A \times B)\) to combine the first two logarithms: \(\log_2 (x - 6) + \log_2 (x - 4) = \log_2 ((x - 6)(x - 4))\).

Apply the logarithmic property that \(\log_b A - \log_b B = \log_b \left(\frac{A}{B}\right)\) to combine all logarithms into a single logarithm: \(\log_2 \left( \frac{(x - 6)(x - 4)}{x} \right) = 2\).

Rewrite the equation \(\log_2 \left( \frac{(x - 6)(x - 4)}{x} \right) = 2\) in its exponential form: \(\frac{(x - 6)(x - 4)}{x} = 2^2\).

Solve the resulting algebraic equation for \(x\), then check each solution against the domain restrictions found in step 1 to reject any extraneous solutions.

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 product, quotient, and power rules, is essential for combining or simplifying logarithmic expressions. For example, log_b(A) + log_b(B) = log_b(AB) and log_b(A) - log_b(B) = log_b(A/B). These rules allow the equation to be rewritten in a simpler form to solve for x.

Video consigliato:

5:36

Change of Base Property

Domain of Logarithmic Functions

The domain of a logarithmic function log_b(x) requires that the argument x be positive (x > 0). When solving logarithmic equations, it is crucial to check that the solutions do not make any logarithm's argument zero or negative, as these values are not valid and must be rejected.

Video consigliato:

5:26

Graphs of Logarithmic Functions

Solving Exponential Equations

After applying logarithmic properties, the equation often converts into an exponential form. Solving the resulting polynomial or exponential equation involves algebraic manipulation to find exact solutions. These solutions can then be approximated using a calculator if needed.

Video consigliato:

5:47

Solving Exponential Equations Using Logs

Pratica correlata

Domanda del libro di testo

In Exercises 71–78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log_0.1 17

804

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. 2 log₃(x+4)=log₃ 9 + 2

706

views

Domanda del libro di testo

Find the domain of each logarithmic function. f(x) = log₅(x+4)

1245

views

Domanda del libro di testo

In Exercises 74–79, solve each logarithmic equation. log2 (x+3) + log2 (x-3) =4

1163

views

Domanda del libro di testo

In Exercises 71–78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log₁₄ 87.5