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Chapter 3: Problem 107

A classmate claims that the following are true. (a) \(\ln (u+v)=\ln u+\ln v=\ln (u v)\) (b) \(\ln (u-v)=\ln u-\ln v=\ln \frac{u}{v}\) (c) \((\ln u)^{n}=n(\ln u)=\ln u^{n}\) Discuss how you would demonstrate that these claims are not true.

Short Answer

Expert verified
All three claims made by the classmate are not valid. The correct logarithm rules are: \(\ln (u * v) = \ln u + \ln v\), \(\ln u - \ln v = \ln \frac{u}{v}\), and \(n \ln u = \ln (u^n)\).

Step by step solution

01

Check the validity of the first claim

The first claim states that \(\ln (u+v)=\ln u+\ln v=\ln (u v)\). However, the correct logarithm rule is that, \(\ln (u*v) = \ln u + \ln v\). The proposed identity \(\ln (u + v) = \ln u + \ln v\) is not accurate.
02

Check the validity of the second claim

The second claim is \(\ln (u-v)=\ln u-\ln v=\ln \frac{u}{v}\). This claim is also incorrect. According to logarithm rules, \(\ln u - \ln v = \ln \frac{u}{v}\) is valid, however, \(\ln (u - v) = \ln u - \ln v\) is not correct.
03

Check the validity of the third claim

Lastly, the third claim \((\ln u)^{n}=n(\ln u)=\ln u^{n}\) needs to be checked. However, this claim is also not correct. According to logarithm rules, \(n \ln u = \ln (u^n)\) is valid. But, \((\ln u)^n\) does not equal to \(n \ln u\) and does not equal to \(\ln (u^n)\).

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Most popular questions from this chapter

Is it possible for a logarithmic equation to have more than one extraneous solution? Explain.

In Exercises \(103-106,\) use the change-of-base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utility to graph the ratio. $$f(x)=\log _{1 / 2} x$$

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$2 \ln (x+3)=3$$

Let \(f(x)=\log _{a} x\) and \(g(x)=a^{x},\) where \(a>1\) (a) Let \(a=1.2\) and use a graphing utility to graph the two functions in the same viewing window. What do you observe? Approximate any points of intersection of the two graphs. (b) Determine the value(s) of \(a\) for which the two graphs have one point of intersection. (c) Determine the value(s) of \(a\) for which the two graphs have two points of intersection.

Evaluate \(g(x)=\ln x\) at the indicated value of \(x\) without using a calculator. $$x=e^{5}$$
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