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 31

Capitolo 5, Problema 31

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)

Guida verificata passo dopo passo

Recognize that the expression involves a logarithm of a cube root, which can be rewritten using fractional exponents. Recall that \( \sqrt[3]{a} = a^{1/3} \). So rewrite the expression as \( \log \left( \left( \frac{x}{y} \right)^{1/3} \right) \).

Use the logarithm power rule, which states \( \log(a^b) = b \log(a) \), to bring the fractional exponent \( \frac{1}{3} \) in front of the logarithm. This gives \( \frac{1}{3} \log \left( \frac{x}{y} \right) \).

Apply the logarithm quotient rule, which states \( \log \left( \frac{a}{b} \right) = \log a - \log b \), to expand \( \log \left( \frac{x}{y} \right) \) into \( \log x - \log y \).

Substitute this back into the expression to get \( \frac{1}{3} ( \log x - \log y ) \).

Distribute the \( \frac{1}{3} \) across the terms inside the parentheses to write the fully expanded form as \( \frac{1}{3} \log x - \frac{1}{3} \log y \).

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

Properties of logarithms include rules such as the product, quotient, and power rules. These allow us to rewrite logarithmic expressions by expanding or condensing them. For example, log(a/b) = log(a) - log(b) and log(a^n) = n·log(a). These properties are essential for simplifying and expanding logarithmic expressions.

Video consigliato:

5:36

Change of Base Property

Radicals and Exponents

Radicals can be expressed as fractional exponents, where the nth root of a number is the same as raising it to the power of 1/n. For example, the cube root of x is x^(1/3). Converting radicals to exponents helps apply logarithm power rules effectively when expanding expressions.

Video consigliato:

04:06

Rational Exponents

Logarithmic Expression Expansion

Expanding logarithmic expressions involves breaking down complex logs into sums, differences, or multiples of simpler logs using the properties of logarithms. This process simplifies expressions and can sometimes allow evaluation without a calculator by recognizing standard log values or simplifying terms.

Video consigliato:

7:30

Logarithms Introduction

Pratica correlata

Domanda del libro di testo

Begin by graphing f(x) = 2^x. 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) = 2.2^x

755

views

Domanda del libro di testo

Graph f(x) = 2^x and g(x) = log2 x in the same rectangular coordinate system. Use the graphs to determine each function's domain and range.

1634

views

Domanda del libro di testo

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

795

views

Domanda del libro di testo

823

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. 5e^x=23

711

views

Domanda del libro di testo

841

views