1. ## log problems

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Very simple, and very easy. I am just not sure why they say 10/3 as they answer in the answer book... When I look at previous questions, to me, it would seem to be 3/3, and 1. My mistake or theirs?

2. Originally Posted by Slipery
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Very simple, and very easy. I am just not sure why they say 10/3 as they answer in the answer book... When I look at previous questions, to me, it would seem to be 3/3, and 1. My mistake or theirs?
No, when you multiply two things, you ADD their exponents, not multiply them.

$\displaystyle \log_{10}{|10^3 \times 10^{\frac{1}{3}}|}$

$\displaystyle \log_{10}{|10^3|} + \log_{10}{|10^{\frac{1}{3}}|}$

$\displaystyle 3 + \frac{1}{3}$

$\displaystyle \frac{9}{3} + \frac{1}{3} = \frac{10}{3}$

3. Ahhhh, no wonder. Thats a pretty absent-minded mistake by myself, Thanks

4. Hello, Slipery!

You forgot one of the rules of logs (or exponents)

$\displaystyle \log\left(1000\sqrt[3]{10}\right)$

We have: .$\displaystyle \log\left(1000\sqrt[3]{10}\right) \:=\:\log\left(10^3\cdot10^{\frac{1}{3}}\right)$ .$\displaystyle = \;\log\left(10^{3+\frac{1}{3}}\right) \;=\;\log\left(10^{\frac{10}{3}}\right) \quad\hdots\;\text{ etc.}$

We have: .$\displaystyle \log\left(10^3\cdot10{\frac{1}{3}}\right) \;=\;\log\left(10^3\right ) + \log\left(10^{\frac{1}{3}}\right) \quad\hdots\;\text{ etc.}$

Got it?