Skip to main content
Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblema 31
Capitolo 5, Problema 31

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 3e5x=1977

Guida verificata passo dopo passo
1
Start with the given equation: \(3e^{5x} = 1977\).
Isolate the exponential term by dividing both sides of the equation by 3: \(e^{5x} = \frac{1977}{3}\).
Apply the natural logarithm (ln) to both sides to undo the exponential function: \(\ln\left(e^{5x}\right) = \ln\left(\frac{1977}{3}\right)\).
Use the logarithmic identity \(\ln\left(e^{a}\right) = a\) to simplify the left side: \(5x = \ln\left(\frac{1977}{3}\right)\).
Solve for \(x\) by dividing both sides by 5: \(x = \frac{1}{5} \ln\left(\frac{1977}{3}\right)\).

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:
2m

Concetti chiave

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

Exponential Equations

An exponential equation is one in which the variable appears in the exponent. Solving such equations often involves isolating the exponential expression and then applying logarithms to both sides to solve for the variable.
Video consigliato:
5:47
Solving Exponential Equations Using Logs

Logarithms and Their Properties

Logarithms are the inverse operations of exponentials. They allow us to solve equations where the variable is an exponent by converting the exponential form into a logarithmic form, using properties like log(a^b) = b log(a).
Video consigliato:
5:36
Change of Base Property

Using Calculators for Approximations

After expressing the solution in logarithmic form, calculators are used to find decimal approximations. This involves evaluating natural (ln) or common (log) logarithms and rounding the result to the desired decimal places.
Video consigliato:
5:47
Solving Exponential Equations Using Logs
Pratica correlata
Domanda del libro di testo

Begin by graphing f(x) = 2x. Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. g(x) = 2.2x

755
views
Domanda del libro di testo

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. logb(xy3z3)\(\log\)_{b}\(\left\)(\(\frac{\sqrt{x}\)y^3}{z^3}\(\right\))

794
views
Domanda del libro di testo

Evaluate each expression without using a calculator. log2 (1/√2)

795
views
Domanda del libro di testo

Begin by graphing f(x) = 2x. Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. g(x) = −2x

823
views
Domanda del libro di testo

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log ∛(x/y)

839
views
Domanda del libro di testo

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. e(1−5x)=793

724
views