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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 73
Capitolo 5, Problema 73

In Exercises 71–78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log14 87.5

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Identify the logarithm you need to evaluate: \( \log_{14} 87.5 \), which means the logarithm of 87.5 with base 14.
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): \( \log_{14} 87.5 = \frac{\log_{10} 87.5}{\log_{10} 14} \).
Use a calculator to find the values of \( \log_{10} 87.5 \) and \( \log_{10} 14 \) separately, making sure to keep at least four decimal places.
Divide the value of \( \log_{10} 87.5 \) by the value of \( \log_{10} 14 \) to get the final result for \( \log_{14} 87.5 \).

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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 14 of 87.5 means finding the exponent x such that 14^x = 87.5. 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 evaluation of logarithms with any base using a calculator.
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5:36
Change of Base Property

Using a Calculator for Logarithms

Calculators can compute common logarithms (log base 10) and natural logarithms (log base e). To find log base 14 of 87.5, use the change of base formula with either log or ln functions on the calculator, then round the result to four decimal places as required.
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7:30
Logarithms Introduction
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