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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 77
Chapter 5, Problem 77

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. logπ 63

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Identify the logarithm expression given: \(\log_{\pi} 63\), which means the logarithm of 63 with base \(\pi\).
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): \(\log_{\pi} 63 = \frac{\log_{10} 63}{\log_{10} \pi}\).
Use a calculator to find the values of \(\log_{10} 63\) and \(\log_{10} \pi\) separately, making sure to keep at least four decimal places.
Divide the two logarithm values obtained to get the value of \(\log_{\pi} 63\), 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? Common logarithms use base 10, while natural logarithms use base e (~2.718). Understanding the base is crucial for evaluating or converting logarithmic expressions.
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Change of Base Formula

The change of base formula allows you to evaluate logarithms with any base using common or natural logarithms: log_b(a) = log_c(a) / log_c(b), where c is a convenient base like 10 or e. This formula is essential when calculators only provide log base 10 or e.
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Change of Base Property

Using a Calculator for Logarithms

Calculators typically have buttons for common logarithms (log base 10) and natural logarithms (ln). To evaluate logs with other bases, use the change of base formula and these functions, then round the result to the required decimal places.
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Logarithms Introduction
Related Practice
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. log(x+4)=log x+log 4

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Textbook Question

Find the domain of each logarithmic function. f(x) = ln (x-2)²

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Textbook Question

Use a graphing utility and the change-of-base property to graph each function. y = log3 x

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Textbook Question

In Exercises 74–79, solve each logarithmic equation. log2 (x+3) + log2 (x-3) =4

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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. log(3x−3)=log(x+1)+log 4

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Textbook Question

Find the domain of each logarithmic function. f(x) = log (2 - x)

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