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Chapter 3: Problem 65

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. $$\frac{1}{2}\left(\log _{5} x+\log _{5} y\right)-2 \log _{5}(x+1)$$

Short Answer

Expert verified
The condensed form of the given logarithmic expression is \(\log_{5}\left(\frac{(xy)^{1/2}}{(x+1)^2}\right)\)

Step by step solution

01

Apply the first logarithm property

To begin with, apply the logarithm property \(\log_b mn = \log_b m + \log_b n\) to the sum of the logarithms, which means replacing \(\log _{5} x+\log _{5} y\) with \(\log_{5} xy\). This gives: \[\frac{1}{2}\log _{5} xy - 2 \log _{5}(x+1)\]
02

Apply the second logarithm property

Next, apply the logarithm property \(\log_b m^n = n \log_b m\) to simplify the coefficients in front of the logarithms. Change \(\frac{1}{2}\log _{5}xy\) to \(\log _{5}(xy)^{1/2}\) and \(2\log_5(x+1)\) to \(\log_5(x+1)^2\), which yields: \[\log _{5}(xy)^{1/2} - \log _{5}(x+1)^2\]
03

Apply the first logarithm property again

Again apply the logarithm property \(\log_b mn = \log_b m + \log_b n\), but this time in reverse due to the subtraction between two logarithmic terms. Thus, replace \(\log _{5}(xy)^{1/2} - \log _{5}(x+1)^2\) with \(\log_{5}\left(\frac{(xy)^{1/2}}{(x+1)^2}\right)\) . The logarithmic expression is simplified to: \[\log_{5}\left(\frac{(xy)^{1/2}}{(x+1)^2}\right)\]

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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.

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This will help you prepare for the material covered in the first section of the next chapter. $$\text { Solve: } \frac{5 \pi}{4}=2 \pi x$$

a. Evaluate: \(\log _{2} 16\) b. Evaluate: \(\log _{2} 32-\log _{2} 2\) c. What can you conclude about \(\log _{2} 16,\) or \(\log _{2}\left(\frac{32}{2}\right) ?\)

Graph \(f\) and \(g\) in the same rectangular coordinate system. Then find the point of intersection of the two graphs. $$f(x)=2^{x}, g(x)=2^{-x}$$
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