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#### Alaba27

##### New member
If log[a]x=5 and log[a]y=8, solve:

log[a]((ax2)/(√y))-2

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I am completely lost. I've tried some ways of doing this question but I can't get past the second and third steps. This is one of the last questions in my homework and I do not have a step-by-step solutions manual, only the final answer which would be useless because I will have no idea how to get there. Can someone please give me a step-by-step solution? Please and thanks!

#### Klaas van Aarsen

##### MHB Seeker
Staff member
If log[a]x=5 and log[a]y=8, solve:

log[a]((ax2)/(√y))-2

---------

I am completely lost. I've tried some ways of doing this question but I can't get past the second and third steps. This is one of the last questions in my homework and I do not have a step-by-step solutions manual, only the final answer which would be useless because I will have no idea how to get there. Can someone please give me a step-by-step solution? Please and thanks!
Welcome to MHB, Alaba27! There are a couple of calculation rules for logarithms.

In particular:
$$\log_a p^q = q \log_a p \\ \log_a pq = \log_a p + \log_a q \\ \log_a \frac p q = \log_a p - \log_a q \\ \sqrt{p} = p^{1/2}$$
Can you apply those?

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
You might need to use :

$$\displaystyle \log_a a = 1$$

#### Alaba27

##### New member
I got the solution! After multiple attempts and help from others I got this:

= (ax2/y1/2)-2
= (a-2[x2]-2)/([y1/2]-2
= (a-2x-4)/(y-1)
= y/a2x4

loga(y/a2x4) = -2[loga(a) + 2loga(x) – 1/2loga(y)]

= -2 -4loga(x) + loga(y)
= -2 – 4(5) + 8
= -2 – 20 + 8
= -14

#### MarkFL

Yes, good work! For the benefit of other students who may read this topic, I will write out a solution method using $\LaTeX$:
If $$\displaystyle \log_a(x)=5$$ and $$\displaystyle \log_a(y)=8$$, find the value of $$\displaystyle \log_a\left(\left(\frac{ax^2}{\sqrt{y}} \right)^{-2} \right)$$.
$$\displaystyle \log_a\left(\left(\frac{ax^2}{\sqrt{y}} \right)^{-2} \right)=-2\log_a\left(\frac{ax^2}{\sqrt{y}} \right)=$$
$$\displaystyle -2\left(\log_a(ax^2)-\log_a(\sqrt{y}) \right)=-2\left(\log_a(a)+\log_a(x^2)-\log_a(y^{\frac{1}{2}}) \right)=$$
$$\displaystyle -2\left(1+2\log_a(x)-\frac{1}{2}\log_a(y) \right)=-2\left(1+2\cdot5-\frac{1}{2}\cdot8 \right)=-2\left(1+10-4 \right)=-2(7)=-14$$