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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 71
Chapter 5, Problem 71

In Exercises 71–78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log5 13

Verified step by step guidance
1
Recognize that the logarithm given is \( \log_5 13 \), which means the logarithm of 13 with base 5.
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): \( \log_5 13 = \frac{\log_{10} 13}{\log_{10} 5} \).
Use a calculator to find the values of \( \log_{10} 13 \) and \( \log_{10} 5 \) separately, making sure to keep enough decimal places for accuracy.
Divide the value of \( \log_{10} 13 \) by \( \log_{10} 5 \) to get \( \log_5 13 \), then round your answer 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 5 of 13 means finding the exponent x such that 5^x = 13. Understanding the relationship between exponents and logarithms is fundamental.
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Change of Base Formula

The change of base formula allows you to evaluate logarithms with any base using common (base 10) or natural (base e) logarithms: log_b(a) = log(a) / log(b). This is essential when calculators only provide log or ln functions, enabling calculation of log base 5 of 13.
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Using a Calculator for Logarithms

Calculators typically have buttons for common logarithms (log base 10) and natural logarithms (ln). To find log base 5 of 13, use the change of base formula with either log or ln, then compute the values and divide, rounding the result to four decimal places as required.
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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. log2(x+2)−log2(x−5)=3

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. 2 log3(x+4)=log3 9 + 2

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. log3(x+6)+log3(x+4)=1

Textbook Question

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

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. logx+log(x2−1)−log7−log(x+1)\(\log\) x + \(\log\)(x^2 - 1) - \(\log\) 7 - \(\log\)(x + 1)

Textbook Question

The figure shows the graph of f(x) = ln x. In Exercises 65–74, 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) = 2 ln x