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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 75
Capitolo 5, Problema 75

In Exercises 71–78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log0.1 17

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Recognize that the logarithm given is \( \log_{0.1} 17 \), which means the logarithm of 17 with base 0.1.
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): \( \log_{0.1} 17 = \frac{\log_{10} 17}{\log_{10} 0.1} \).
Use a calculator to find \( \log_{10} 17 \) and \( \log_{10} 0.1 \) separately, keeping the values to at least 5 decimal places for accuracy.
Divide the value of \( \log_{10} 17 \) by \( \log_{10} 0.1 \) to get the final result, then round your answer to four decimal places.

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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 0.1 of 17 means finding the exponent x such that 0.1^x = 17. Understanding how the base affects the logarithm is crucial for solving the problem.
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Logarithms Introduction

Change of Base Formula

The change of base formula allows you to compute logarithms with any base using common (base 10) or natural (base e) logarithms: log_b(a) = log_c(a) / log_c(b). This is essential here because calculators typically only compute log base 10 or e, not base 0.1.
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5:36
Change of Base Property

Using a Calculator for Logarithms

Calculators can evaluate common logarithms (log base 10) and natural logarithms (log base e) directly. By applying the change of base formula, you can use these functions to find logarithms with other bases and round the result to four decimal places as required.
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Logarithms Introduction
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