Domanda del libro di testo
Graph functions f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. f(x) = 3x and g(x) = 3-x
882
views
Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 5, Problema 47
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. 32x+3x−2=0
Guida verificata passo dopo passo1
Start by recognizing that the equation involves exponential expressions with the same base: \(3^{2x} + 3^x - 2 = 0\). Notice that \$3^{2x}$ can be rewritten as $(3^x)^2$.
Introduce a substitution to simplify the equation. Let \(y = 3^x\). Then the equation becomes \(y^2 + y - 2 = 0\).
Solve the quadratic equation \(y^2 + y - 2 = 0\) using the quadratic formula: \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1\), \(b=1\), and \(c=-2\).
After finding the values of \(y\), recall that \(y = 3^x\). Solve for \(x\) by taking the logarithm of both sides: \(x = \log_3(y)\). You can express this using natural logarithms as \(x = \frac{\ln(y)}{\ln(3)}\) or common logarithms as \(x = \frac{\log(y)}{\log(3)}\).
Evaluate the logarithmic expressions using a calculator to find the decimal approximations of \(x\), rounding to two decimal places as required.
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:
3mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Exponential Equations
Exponential equations involve variables in the exponent, such as 3^(2x) or 3^x. Solving these requires rewriting the equation to isolate the exponential expressions or to express them in a common base, enabling the use of algebraic methods to find the variable.
Video consigliato:
5:47
Solving Exponential Equations Using Logs
Logarithms and Their Properties
Logarithms are the inverse operations of exponentials, allowing us to solve for variables in exponents. Natural logarithms (ln) and common logarithms (log) help convert exponential equations into linear forms, making it easier to isolate and solve for the unknown.
Video consigliato:
5:36Change of Base Property
Quadratic Form in Exponential Equations
Some exponential equations can be transformed into quadratic equations by substituting expressions like 3^x = y. This substitution simplifies the problem to solving a quadratic equation, after which the original variable can be found by reversing the substitution.
Video consigliato:
05:35Introduction to Quadratic Equations
Pratica correlata
Domanda del libro di testo
Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. log x + 3 log y
998
views
Domanda del libro di testo
Graph functions f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. f(x) = 3x and g(x) = (1/3). 3x
998
views
Domanda del libro di testo
Graph f(x) = (1/2)x and g(x) = log1/2 x in the same rectangular coordinate system.
1526
views
Domanda del libro di testo
The figure shows the graph of f(x) = ex. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn h(x) = e2x + 1
793
views
Domanda del libro di testo
Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. log2 (96) - log2 (3)
915
views