# Thread: Need help solving this logarithm

1. ## Need help solving this logarithm

Write (equation) as the sum, difference, and/or constant multiple of logarithms.

Equation is attached.

2. Hello,
Originally Posted by mwok
Write (equation) as the sum, difference, and/or constant multiple of logarithms.

Equation is attached.
For any base of the logarithm, you have :
$\displaystyle \log_a (xy)=\log_a(x)+\log_a(y)$
$\displaystyle \log_a \left(\frac xy\right)=\log_a(x)-\log_a(y)$
$\displaystyle \log_a (x^y)=y \log_a(x)$

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Note that $\displaystyle \sqrt{z}=z^{1/2}$
$\displaystyle \log_5 \left(\frac{x^3}{y z^{1/2}}\right)=\log_5(x^3)-\log_5(yz^{1/2})=\dots$

Can you finish it ?

3. Thanks, attached is the solution I came up with (is it correct?).

BTW, does the solution answer all parts of the question (as the sum, difference, and/or constant multiple of logarithms?)

4. Originally Posted by mwok
Thanks, attached is the solution I came up with (is it correct?).

BTW, does the solution answer all parts of the question (as the sum, difference, and/or constant multiple of logarithms?)
Hey it's a z, not a 2 !

And it's not a sum and difference of logarithms...

Here is the thing :

$\displaystyle \log_5 \left(\frac{x^3}{y z^{1/2}}\right)=\log_5(x^3)-\log_5(yz^{1/2})$
$\displaystyle =3 \log_5(x) \quad - \quad \left(\log_5(y)+\log_5(z^{1/2})\right)$
$\displaystyle =3 \log_5(x)-\log_5(y)-\frac{\log_5(z)}{2}$

This is what you're asked to find. Do you understand the steps ? And what they mean by "sum, difference, and/or constant multiple of logarithms" ?

5. Thanks! Ah I see the solution and understand the steps.

But what do they mean by "sum, difference, and/or..."?

Does it mean just show the equation as a sum/difference of logarithms? Like in this case, we've shown a difference of logarithms?