Question

Solving an equation involving logarithms:
\[(\log_3 x)^2 = \log_3 x^2 + 3\]
Specifically, I am looking on guidance on what to do with the left hand side. I'm not sure if there are any rules that govern multiplying logs, though I don't think there are.

Answer 1

\[(\log_3x)^2=\log_3(x^2)+3\]is this right?

Answer 2

Yep.

Answer 3

let\[u=\log_3x\]then we have\[u^2=2u+3\]which is a quadratic we can solve for u
from that we can get x

Answer 4

\[\log_{3}x^{2} = 2 \log_{x}3 \]

Therefor:
\[(log_{3}x)^{2} \ = 2 \log_{x}3 + 3\]\[(log_{3}x)^{2} \ - 2 \log_{x}3 - 3=0\]Assume: \[(log_{3}x) \ = a\]So:

a^2 -2a - 3 = 0
(a-3)(a+1)=0

So logx = 3 and logx = -1

Answer 5

Thanks to both of you! That gave me the correct answers of 1/3 and 27.