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Chapter 12: Problem 63

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$\log _{14} 87.5$$

Short Answer

Expert verified
The value of \( \log_{14} 87.5 \), rounded to four decimal places, is to be calculated using the calculator after changing to the base of your preference (10 or e).

Step by step solution

01

Change of Base Formula

The change of base formula for logarithms states that for any two positive real numbers a and b, and for any positive real number n in the base a logarithm, the value of such a logarithm may also be written in terms of logarithms with a base of b. Specifically, \[ \log_{a}n = \frac{\log_{b}n}{\log_{b}a} \]. Applying this formula to the given exercise where a = 14 and n = 87.5, we can convert it to use the base 10 logarithm (common logarithm) or base e logarithm (natural logarithm) because these are readily available in calculators.
02

Evaluation using a calculator

The expression becomes \[ \frac{\log_{10} 87.5}{\log_{10} 14} \] or \[ \frac{\ln 87.5}{\ln 14} \]. Use a calculator to research these expressions and keep in mind that the question asks for a number rounded to four decimal places.
03

Result

Now you finish evaluating in your calculator and round the resulting number to four decimal places.

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Most popular questions from this chapter

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.

Use a graphing utility and the change-of-base property to graph \(y=\log _{3} x, y=\log _{25} x,\) and \(y=\log _{100} x\) in the same viewing rectangle. a. Which graph is on the top in the interval \((0,1) ?\) Which is on the bottom? b. Which graph is on the top in the interval \((1, \infty) ?\) Which is on the bottom? c. Generalize by writing a statement about which graph is on top, which is on the bottom, and in which intervals, using \(y=\log _{b} x\) where \(b>1\)

Students in a mathematics class took a final examination. They took equivalent forms of the exam in monthly intervals thereafter. The average score, \(f(t),\) for the group after \(t\) months is modeled by the human memory function \(f(t)=75-10 \log (t+1),\) where \(0 \leq t \leq 12\) Use a graphing utility to graph the function. Then determine how many months will elapse before the average score falls below 65

Solve: $$\frac{3}{x+1}-\frac{5}{x}=\frac{19}{x^{2}+x}$$

Solve each equation. $$\log _{2}(x-6)+\log _{2}(x-4)-\log _{2} x=2$$
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