Logs

**Jimgotkp** · May 31st, 2007, 04:33 PM

Directions: use the properties of logarithms to write the expression as a sum, difference, or multiples of logarithms of x, y, or constant.

log (4th root of (x/y))

thanks for your help. i appreciate it

**ThePerfectHacker** · May 31st, 2007, 04:41 PM

Originally Posted by Jimgotkp

Directions: use the properties of logarithms to write the expression as a sum, difference, or multiples of logarithms of x, y, or constant.

log (4th root of (x/y))

thanks for your help. i appreciate it

$\log \sqrt[4]{\frac{x}{y}} = \log \left(\frac{x}{y}\right)^{1/4} = \frac{1}{4} \log \frac{x}{y} = \frac{1}{4}\log x - \frac{1}{4}\log y$

**Jimgotkp** · May 31st, 2007, 04:43 PM

wow thanks for those fast replies. i was trying to ask my friends in ap calc, and they totally forgot how to do it.

THANK YOU!

have a math quiz on this stuff tomorrow

**Krizalid** · May 31st, 2007, 04:50 PM

But post all exercises that you can't solve.

**Jimgotkp** · May 31st, 2007, 05:02 PM

okay here is a list of log problems so that i can check it later tonight.

Use the properties of logarithms to write the expression as a sum, difference, or multiples of logarithms of x, y, constants.

1. log1000x4 <-(x to the 4th. i dont have those programs to write perfect math problems, sorry.)
2. In(3rd root of x) / (3rd root of y)
3. 1/3logx
4. 4log(x-1) + 2log(x+4)
5. 1/2log a(x-3) - 1/3log a(x+3)
6. 1/4log b(x+2) - 1/5log b(3-x)

**Jonboy** · May 31st, 2007, 05:44 PM

Hi Jimgotkp! I helped with 2,3,4 and half of 5. You all check and make sure I didn't make any mistakes because I have to get off the comp.

Use the properties of logarithms to write the expression as a sum, difference, or multiples of logarithms of x, y, constants.

2. $\ln \frac{{\sqrt[3]{x}}}{{\sqrt[3]{y}}}$

Since $log_{b}a\,-\,log_{b}c\,=\,log_{b}\frac{a}{c}$ , Problem # 2 can be rewritten as $ln{\sqrt[3]{x}}\,-\,ln{\sqrt[3]{y}}$

3. $\frac{1}{3}logx$

A rule says: $x\,log_{b}\,a\,=\,log_{b}\,a^{x}$

so you can rewrite this is as: $log\,x^{\frac{1}{3}}$ . I don't see how you can simplify this further.

4. $4\,log(x\,-\,1)\,+\,2\,log(\,x\,+\,4)$

Using the previous rule, we can rewrite this as: $log(x\,-\,1)^{4}\,+\,log(\,x\,+\,4)^{2}$

Then as a multiple: $\log [(x - 1)^4 (x + 4)^2]$

5. $\frac{1}{2}log_{a}(x\,-\,3)\,-\,\frac{1}{3}log_{a}(x\,+\,3)$

Move the fraction: $log_{a}(x\,-\,3)^{\frac{1}{2}}\,-\,log_{a}(x\,+\,3)^{\frac{1}{3}}$
Once again,.. $log_{b}a\,-\,log_{b}c\,=\,log_{b}\frac{a}{c}$ : .... $log_{a}\frac{(x\,-\,3)^{\frac{1}{2}}}{(x\,+\,3)^{\frac{1}{3}}}$

**Jimgotkp** · May 31st, 2007, 05:55 PM

sorry moderator, i didnt see the rules before.

jonboy, thank you very much. im just trying to clarify the answers up.

**Jhevon** · May 31st, 2007, 06:03 PM

Originally Posted by Jimgotkp

sorry moderator, i didnt see the rules before.

lol, i'm not a moderator!

are you sure these are the instructions for all the questions? some of these are already in the form you are asking us to put them in, it would be a better exercise to combine them rather than express them as sums and whatnot. i'll do the ones Jonboy left out.

Originally Posted by Jimgotkp

Use the properties of logarithms to write the expression as a sum, difference, or multiples of logarithms of x, y, constants.

1. log1000x4 <-(x to the 4th. i dont have those programs to write perfect math problems, sorry.)

$\log 1000x^4 = \log 1000 + \log x^4 = 3 + 4 \log x$ .......i assumed we are dealing with log to the base 10 here, so that's why i got the 3

6. 1/4log b(x+2) - 1/5log b(3-x)

$\frac {1}{4} \log_{b} (x + 2) - \frac {1}{5} \log_{b} (3 - x) = \log_{b} (x + 2)^{ \frac {1}{4}} - \log_{b} (3 - x)^{ \frac {1}{5}}$

$= \log_{b} \sqrt [4] {x + 2} - \log_{b} \sqrt [5] {3 - x}$

$= \log_{b} \left( \sqrt [4] {x + 2} \sqrt [5] {3 - x} \right)$

if you have any questions, please feel free to ask

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