If $\log_a3=p$ and $\log_a2=q$ what strategies deduce an expression for $\log_a(4.5a^2)$?

Question

I've considered the exponent forms $a^p=3$, $a^q=2$ and $a^x=4.5a^2$ and other qualities of logs such as $\log_aa=1$ but I don't see how to begin with this one. I've also considered the additive, subtractive and exponent log laws.

Is it possible to go further than merely saying: $$p(4.5q)$$ $$4.5pq$$

Also, I can use log law 3 to say that $2log_a4.5a$... this seems to me to be heading towards some kind of solution. — duckegg, Jun 28 '22 at 21:21
@duckegg Careful: the whole expression isn't squared, just the $a,$ so you can't quite do that. Instead I would recommend doing $\log_a 4.5a^2 = \log_a 4.5 + \log_a a^2 = \log_a 4.5 + 2 \log_a a$ to apply this rule — Stephen Donovan, Jun 28 '22 at 21:26

score 1 · Accepted Answer · answered Jun 28 '22 at 21:41

1

Assuming a > 0, you can do the following:

$$ \begin{align*} log_a(4.5a^2) &= log_a(4.5) + log_a(a^2) \\ &= log_a(4.5) + log_a(a^2) \\ &= log_a\left(\frac{3 . 3}{2}\right) + 2.log_a(a) \\ &= log_a(3^2) - log_a(2) + 2 \\ &= 2.log_a(3) - log_a(2) + 2 \\ &= 2p - q + 2 \\ \end{align*} $$

answered Jun 28 '22 at 21:41

babbupandey

285

wow... yes, I see what you did there. You used all of the techniques the course I am in has covered so far but you did so in a creative way that had not occurred to me. – duckegg Jun 28 '22 at 21:49
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Thank you! If you think this answers the question then you may mark it as an accepted answer for posterity. – babbupandey Jun 28 '22 at 21:53
Done :). I read your profile and your work sounds interesting by the way. Let me know about that final script you mentioned, I could use it too haha – duckegg Jun 28 '22 at 21:56
Which final script? – babbupandey Jun 28 '22 at 21:57
Your 'About': final paragraph, last sentence. – duckegg Jun 28 '22 at 21:59
Oh, hahahahaha!! Wrote it long time back – babbupandey Jun 28 '22 at 22:01

If $\log_a3=p$ and $\log_a2=q$ what strategies deduce an expression for $\log_a(4.5a^2)$?

1 Answers1