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Please help with this logarithm problem!

Alaba27

New member
Apr 4, 2013
18
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!
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,780
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
Jan 17, 2013
1,667
You might need to use :

\(\displaystyle \log_a a = 1\)
 

Alaba27

New member
Apr 4, 2013
18
I just don't understand how to use those formulas with this kind of the question. None of the other questions in my homework are in that format and it's extremely confusing. This is what it looks like.

a.png
 

Alaba27

New member
Apr 4, 2013
18
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

Administrator
Staff member
Feb 24, 2012
13,775
Yes, good work! (Yes)

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 \(\displaystyle \log_a\left(\left(\frac{ax^2}{\sqrt{y}} \right)^{-2} \right)\).

\(\displaystyle \log_a\left(\left(\frac{ax^2}{\sqrt{y}} \right)^{-2} \right)=-2\log_a\left(\frac{ax^2}{\sqrt{y}} \right)=\)

\(\displaystyle -2\left(\log_a(ax^2)-\log_a(\sqrt{y}) \right)=-2\left(\log_a(a)+\log_a(x^2)-\log_a(y^{\frac{1}{2}}) \right)=\)

\(\displaystyle -2\left(1+2\log_a(x)-\frac{1}{2}\log_a(y) \right)=-2\left(1+2\cdot5-\frac{1}{2}\cdot8 \right)=-2\left(1+10-4 \right)=-2(7)=-14\)