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Chapter 3: Problem 72

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 \(\log_{6} 17\) rounded to four decimal places is approximated using \(\frac{\log 17}{\log 6}\) or \(\frac{\ln 17}{\ln 6}\).

Step by step solution

01

Apply the Change of Base Formula

Use the change of base formula to rewrite the expression in terms of common logarithms or natural logarithms. So, \(\log_{6} 17\) would be rewritten as \(\frac{\log 17}{\log 6}\) or \(\frac{\ln 17}{\ln 6}\).
02

Evaluate the Numerator and Denominator

Using a scientific calculator, calculate the values of \(\log 17\) or \(\ln 17\) for the numerator, and \(\log 6\) or \(\ln 6\) for the denominator.
03

Divide the Numerator by the Denominator

Divide the value computed in the numerator by the denominator.
04

Round to Four Decimal Places

The final answer should be rounded to four decimal places given in the exercise instruction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Logarithms
Common logarithms are logarithms with base 10, commonly denoted as \(\log\). They are frequently used in science and engineering due to their simplicity.
Here's why they are useful:
  • They simplify calculations by reducing the range of numbers.
  • They convert multiplication into addition, which is easier to handle.
In the context of our exercise, when you have \(\log_{6} 17\), you can apply the change of base formula. This allows you to convert the base to 10, easily handled by most scientific calculators.
By rewriting as \(\frac{\log 17}{\log 6}\), you can evaluate it efficiently. Remember, common logarithms use base 10, making them straightforward and widely applicable.
Natural Logarithms
Natural logarithms use the base \(e\), an important constant approximately equal to 2.718. It is written as \(\ln\). Natural logarithms are prevalent in growth and decay models, like population growth and radioactive decay.
Why use natural logarithms?
  • They appear naturally in calculus and are integral to exponential functions.
  • They have elegant properties that simplify complex equations.
For the problem \(\log_{6} 17\), you can also use the change of base formula with natural logarithms: \(\frac{\ln 17}{\ln 6}\). The choice between natural and common logarithms can depend on the specific context or your preference, as both will give the same result here.
Scientific Calculator
A scientific calculator is an essential tool for evaluating logarithms, especially when dealing with non-standard bases. Here's how it helps:
  • It has dedicated buttons for \(\log\) and \(\ln\), making calculations swift.
  • It provides precision, crucial for rounding answers to four decimal places.
When you calculate \(\log 17\) or \(\ln 17\) and \(\log 6\) or \(\ln 6\), a scientific calculator ensures accuracy.
This precision is vital for performing the division step and achieving an accurate result. After computing the values, remember to round your answer to four decimal places, as instructed in the exercise. A handy calculator not only speeds up this process but also ensures accuracy.

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Most popular questions from this chapter

This group exercise involves exploring the way we grow. Group members should create a graph for the function that models the percentage of adult height attained by a boy who is \(x\) years old, \(f(x)=29+48.8 \log (x+1) .\) Let \(x=5,6\) 7....., \(15,\) find function values, and connect the resulting points with a smooth curve. Then create a graph for the function that models the percentage of adult height attained by a girl who is \(x\) years old, \(g(x)=62+35 \log (x-4)\) Let \(x=5,6,7, \ldots, 15,\) find function values, and connect the resulting points with a smooth curve. Group members should then discuss similarities and differences in the growth patterns for boys and girls based on the graphs.

The loudness level of a sound, \(D,\) in decibels, is given by the formula $$ D=10 \log \left(10^{12} I\right) $$ where \(I\) is the intensity of the sound, in watts per meter.\(^{2} .\) Decibel levels range from \(0,\) a barely audible sound, to \(160,\) a sound resulting in a ruptured eardrum. (Any exposure to sounds of I30 decibels or higher puts a person at immediate risk for hearing damage.) Use the formula to solve What is the decibel level of a normal conversation, \(3.2 \times 10^{-6}\) watt per meter \(^{2} ?\)

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. $$\log _{3} 7=\frac{1}{\log _{7} 3}$$

Will help you prepare for the material covered in the next section. The formula \(A=10 e^{-0.003 t}\) models the population of Hungary, \(A\), in millions, \(t\) years after 2006 . a. Find Hungary's population, in millions, for \(2006,2007\), \(2008,\) and \(2009 .\) Round to two decimal places. b. Is Hungary's population increasing or decreasing?

Find \(\ln 2\) using a calculator. Then calculate each of the following: \(1-\frac{1}{2} ; \quad 1-\frac{1}{2}+\frac{1}{3} ; \quad 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}\) \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5} ; \ldots .\) Describe what you observe.
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