Logarithms, exponents, and radicals

Hello, kwinnlexa!

3. Find the error of in this reasoning.
State where the erroneous reasoning occured, as well as why it is wrong:

\(\displaystyle \text{Given: }\;\quad\;\left(\tfrac{1}{2}\right)^3 \;>\:\left(\tfrac{1}{2}\right)^4\)

\(\displaystyle \text{Step 1:}\;\;\;\ln\left(\tfrac{1}{2}\right)^3 \;>\; \ln\left(\tfrac{1}{2}\right)^4\)

\(\displaystyle \text{Step 2: }\;3\ln\left(\tfrac{1}{2}\right)\; > \;4\ln\left(\tfrac{1}{2}\right)\)
. . . . . . . . . . . . . . . . . . . Here!
\(\displaystyle \text{Step 3: }\;\qquad\;\; 3 \;>\; 4\)
Click to expand...

\(\displaystyle \text{Going from Step 2 to Step 3, they divided by }\ln\left(\tfrac{1}{2}\right)\;\;\hdots \;\text{ a }negative\text{ quamtity}\)

\(\displaystyle \text{The inequality must be reversed.}\)

 
Do you have your instructor's permission to have other people work on your test with you?

I'm thinking that you should probably do as much work on your own as possible.

On the first test question, the given expression under the root sign (the radicand) is a product of factors. There is a property for taking roots that says we can break the fifth root of the radicand into a product of fith roots of the factors.

\(\displaystyle \sqrt[5]{192 \cdot x^6 \cdot y^5 \cdot z} \;=\; \sqrt[5]{192} \cdot \sqrt[5]{x^6} \cdot \sqrt[5]{y^5} \cdot \sqrt[5]{z}\)

Just simplify each factor.

For the second test question, look up the terms "half life" or "exponential decay" in the index of your textbook. Let us know what you don't understand about the examples.

For the third test question, what is the value of ln(1/2)? What does this value have to do with multiplying inequalities?

Is this a placement exam practice test or is it from an actual class?
 
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