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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblem 59
Chapter 5, Problem 59

In Exercises 58–59, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log4 0.863

Verified step by step guidance
1
Recognize that the problem asks for the logarithm of 0.863 with base 4, written as \(\log_{4} 0.863\).
Recall the change of base formula for logarithms: \(\log_{a} b = \frac{\log_{c} b}{\log_{c} a}\), where \(c\) can be any positive number (commonly 10 or \(e\)).
Apply the change of base formula using common logarithms (base 10): \(\log_{4} 0.863 = \frac{\log_{10} 0.863}{\log_{10} 4}\).
Use a calculator to find the values of \(\log_{10} 0.863\) and \(\log_{10} 4\) separately, making sure to keep the values to at least four decimal places.
Divide the two logarithm values obtained in the previous step to get the value of \(\log_{4} 0.863\), rounded to four decimal places.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Logarithms and Their Bases

A logarithm answers the question: to what power must the base be raised to produce a given number? In this problem, log base 4 of 0.863 means finding the exponent x such that 4^x = 0.863. Understanding the relationship between exponents and logarithms is fundamental.
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Logarithms Introduction

Change of Base Formula

Since calculators typically only compute logarithms with base 10 (common logs) or base e (natural logs), the change of base formula allows conversion: log_b(a) = log_c(a) / log_c(b), where c is 10 or e. This formula enables evaluating log base 4 of 0.863 using a calculator.
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Change of Base Property

Using a Calculator for Logarithms

Calculators can compute common logarithms (log base 10) and natural logarithms (log base e). By applying the change of base formula, you input log(0.863) divided by log(4) or ln(0.863) divided by ln(4) to find the value. Rounding the result to four decimal places completes the evaluation.
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Related Practice
Textbook Question

In Exercises 53-58, begin by graphing f(x) = log₂ x. Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. g(x) = (1/2)log₂ x

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Textbook Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. (1/2)ln x - ln y

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Textbook Question

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. 4 ln (x + 6) - 3 ln x

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Textbook Question

Graph y= 2x and x = 2y in the same rectangular coordinate system.

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Textbook Question

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. log4(3x+2)=3

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Textbook Question

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. g(x) = log(x − 1)

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