Domanda del libro di testo
Give the equation of each exponential function whose graph is shown.
944
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 62
In Exercises 60–63, 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. (log2 x)^4 = 4 log2 x
Guida verificata passo dopo passo1
Step 1: Recall the logarithmic property that states \( a \cdot \log_b(x) = \log_b(x^a) \). This property allows us to manipulate logarithmic expressions involving exponents.
Step 2: Analyze the left-hand side of the equation \((\log_2(x))^4\). This expression means that the logarithm \(\log_2(x)\) is raised to the fourth power, which is different from multiplying the logarithm by 4.
Step 3: Compare this to the right-hand side of the equation \(4 \cdot \log_2(x)\). This expression represents the logarithm \(\log_2(x)\) being multiplied by 4, not raised to the fourth power.
Step 4: Conclude that \((\log_2(x))^4 \neq 4 \cdot \log_2(x)\) because raising a logarithm to a power is not the same as multiplying it by a constant. Therefore, the given equation is false.
Step 5: To make the statement true, rewrite the left-hand side to match the right-hand side. For example, if the equation were \(4 \cdot \log_2(x) = 4 \cdot \log_2(x)\), it would be true. Alternatively, if the left-hand side were \((\log_2(x))^4\), the right-hand side would need to be \((\log_2(x))^4\) to make the equation true.
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:
1mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Properties of Logarithms
Understanding the properties of logarithms is essential for manipulating logarithmic expressions. Key properties include the product, quotient, and power rules, which allow us to simplify or expand logarithmic terms. For instance, the power rule states that log_b(a^n) = n * log_b(a), which is crucial for solving equations involving logarithms.
Video consigliato:
5:36
Change of Base Property
Exponential and Logarithmic Equivalence
Logarithms and exponents are inversely related, meaning that if b^y = x, then log_b(x) = y. This relationship is fundamental in solving logarithmic equations, as it allows us to convert between exponential and logarithmic forms. Recognizing this equivalence helps in verifying the truth of logarithmic statements.
Video consigliato:
7:30Logarithms Introduction
Equation Verification
To determine if an equation is true or false, one must verify both sides of the equation under the same conditions. This often involves substituting values or simplifying expressions. If the two sides do not match, adjustments must be made to create a true statement, which may involve applying logarithmic properties or algebraic manipulation.
Video consigliato:
06:00Categorizing Linear Equations
Pratica correlata
Domanda del libro di testo
The figure shows the graph of f(x) = log x. In Exercises 59–64, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. h(x) = log x − 1
577
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. 6+2 ln x=5
923
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. (1/2)(log x + log y)
794
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. 3 ln x + 5 ln y - 6 ln z
1026
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. 5 ln(2x)=20
809
views