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 22
Capitolo 5, Problema 22

In Exercises 19–29, evaluate each expression without using a calculator. If evaluation is not possible, state the reason. log16 4

Guida verificata passo dopo passo
1
Rewrite the logarithmic expression using the definition of a logarithm: \( \log_b a = c \) means \( b^c = a \). Here, \( \log_{16} 4 \) implies finding the value of \( c \) such that \( 16^c = 4 \).
Express the base (16) and the result (4) as powers of the same base. Note that \( 16 = 2^4 \) and \( 4 = 2^2 \).
Substitute these expressions into the equation \( 16^c = 4 \). This becomes \( (2^4)^c = 2^2 \).
Simplify the left-hand side using the power rule \( (a^m)^n = a^{m \cdot n} \). This gives \( 2^{4c} = 2^2 \).
Since the bases are the same, set the exponents equal to each other: \( 4c = 2 \). Solve for \( c \) by dividing both sides by 4: \( c = \frac{2}{4} \).

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.

Logarithms

A logarithm is the inverse operation to exponentiation, representing the power to which a base must be raised to obtain a given number. For example, in the expression log_b(a), b is the base, and the result is the exponent x such that b^x = a. Understanding logarithms is essential for evaluating expressions like log16 4.
Video consigliato:
7:30
Logarithms Introduction

Change of Base Formula

The change of base formula allows us to convert logarithms from one base to another, making them easier to evaluate. It states that log_b(a) can be expressed as log_k(a) / log_k(b) for any positive k. This is particularly useful when the base is not a common one, such as in log16 4, where we can convert to base 2 or 10.
Video consigliato:
5:36
Change of Base Property

Properties of Logarithms

Logarithms have several properties that simplify their evaluation, including the product, quotient, and power rules. For instance, log_b(mn) = log_b(m) + log_b(n) and log_b(m/n) = log_b(m) - log_b(n). These properties can help break down complex logarithmic expressions into simpler components, aiding in their evaluation.
Video consigliato:
5:36
Change of Base Property
Pratica correlata
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. 10x=3.91

893
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 (x2 y)

821
views
Domanda del libro di testo

Evaluate each expression without using a calculator. log4 16

943
views
Domanda del libro di testo

The graph of an exponential function is given. Select the function for each graph from the following options:

f(x)=3x,g(x)=3x−1,h(x)=3x−1,f(x)=−3x,G(x)=3−x,H(x)=−3−x.f(x) = 3^x, \(\quad\) g(x) = 3^{x-1}, \(\quad\) h(x) = 3^x - 1, \\ f(x) = -3^x, \(\quad\) G(x) = 3^{-x}, \(\quad\) H(x) = -3^{-x}.

1483
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. log4(x64)\(\log\)_4\(\left\)(\(\frac{\sqrt{x}\)}{64}\(\right\))

814
views
Domanda del libro di testo

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. e(x+1)=1/e

745
views