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Chapter 4: Problem 72

In Exercises \(71-78,\) use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{6} 17 $$

Short Answer

Expert verified
The value of \(log_6 17\) to four decimal places, is approximately 1.7654. This uses the change of base formula and a calculator to compute.

Step by step solution

01

Understand the formula to be used - Change of Base Formula

The 'Change of Base Formula' applies to logarithms, which allows changing the base of the logarithm into another number. The formula is \(log_b a = log_c a / log_c b\), where c could be any positive number.
02

Apply the Change of Base Formula to the given problem

Applying the formula to the given logarithm, we need to find \(log_6 17\), but we only have common logarithms (log) or natural logarithms (ln) on the calculator. Let's use the common logarithm (base 10). The formula then changes to: \(log_6 17 = log 17 / log 6\).
03

Calculate the values using a calculator

Calculate the values of \(log 17\) and \(log 6\) using a calculator, and then divide the two outputs.

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Most popular questions from this chapter

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \log _{2} 96-\log _{2} 3 $$

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 7 \ln x-3 \ln y $$

The formula $$ t=\frac{1}{c}[\ln A-\ln (A-N)] $$ describes the time, \(t,\) in weeks, that it takes to achieve mastery of a portion of a task, where \(A\) is the maximum learning possible, \(N\) is the portion of the learning that is to be achieved, and \(c\) is a constant used to measure an individual's learning style. a. Express the formula so that the expression in brackets is written as a single logarithm. b. The formula is also used to determine how long it will take chimpanzees and apes to master a task. For example, a typical chimpanzce learning sign language can master a maximum of 65 signs. Use the form of the formula from part (a) to answer this question: How many weeks will it take a chimpanzee to master 30 signs if \(c\) for that chimp is \(0.03 ?\)

Explain how to use your calculator to find \(\log _{14} 283\).

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 5 \ln x-2 \ln y $$
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