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Chapter 13: Problem 14

Write as the sum or difference of logarithms and simplify, if possible. Assume all variables represent positive real numbers. $$\log _{5} \frac{20}{17}$$

Short Answer

Expert verified
\( \log _{5} \frac{20}{17} = \log _{5}(20) - \log _{5}(17) \)

Step by step solution

01

Identify the quotient property of logarithms

We are given the logarithmic fraction \(\log _{5} \frac{20}{17}\). To rewrite it as a sum or difference of logarithms, we need to use the quotient property of logarithms: \(\log_{a} \left( \frac{x}{y} \right) = \log_{a}x - \log_{a}y\)
02

Apply the quotient property

Now, we are ready to apply the quotient property to the given logarithmic fraction. We can rewrite \(\log _{5} \frac{20}{17}\) as the difference of logarithms. \[ \log _{5} \frac{20}{17} = \log _{5}(20) - \log _{5}(17) \]
03

Simplify the logarithmic expression

Since we cannot simplify \(\log _{5}(20)\) and \(\log _{5}(17)\) further, the given term is already expressed as the sum or difference of logarithms and simplified: \[ \log _{5} \frac{20}{17} = \log _{5}(20) - \log _{5}(17) \]

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Most popular questions from this chapter

Radioactive carbon- 14 is a substance found in all living organisms. After the organism dies, the carbon- 14 decays according to the equation $$y=y_{0} e^{-0.000121 t}$$ where \(t\) is in years, \(y_{0}\) is the initial amount present at time \(t=0,\) and \(y\) is the amount present after \(t\) yr. a) If a sample initially contains 15 g of carbon- 14 how many grams will be present after 2000 yr? b) How long would it take for the initial amount to decay to 10 g? c) What is the half-life of carbon- \(14 ?\)

The hydronium ion concentrations, \(\left[\mathrm{H}^{+}\right],\) are given for some common substances. Find the \(\mathrm{pH}\) of each substance (to the tenths place), and determine whether each substance is acidic or basic. $$\text { Egg white: }\left[\mathrm{H}^{+}\right]=2 \times 10^{-8}$$

Determine whether each function is one-to-one. If it is one-to-one, find its inverse. $$f=\\{(-4,3),(-2,-3),(2,-3),(6,13)\\}$$

If \(f(x)=x+9,\) show that \(f^{-1}(x)=x-9\)

Find the inverse of each one-to-one function. Then, graph the function and its inverse on the same axes. $$h(x)=-\frac{1}{3} x$$
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