Question

Solve for x:
\[(\log_3 x)^3 = 9 \log x\]

Specifically, I'm looking for guidance on how to solve the LHS whatnot with the exponent of three. I don't believe there are any log rules that can simplify that expression.

Answer 1

is that right side log base 10?

Answer 2

Yes.

Answer 3

ok so they're the same log base...

Answer 4

The L.H.S its log base is 3

Answer 5

The LHS is \[\large{(log_3 x)^3}\], with base 3 as @Eyad mentioned.

Answer 6

I got 1 and 27 which seemed to work. However, I'm uncertain how you got the above equation from the original one. Could you divulge in some steps on how you transformed them?

Answer 7

Actually, only 1 is the answer. 27 doesn't work.

Answer 8

Did you use the base change formula?

Answer 9

Sure.

Answer 10

I guess what I'm not sure about is how you got \(9 \log x\) into a log with base 3.

Answer 11

oh.. i'm sorry... i was under the assumption that the right hand side was base 3...
my bad... :(

Answer 12

ok...
maybe you can change that RHS so it has a log base 3 also.. to
\[\large \frac {9\log_3 x}{\log_3 10}\]

Answer 13

I tried this; but it seems that multiplying both sides by log_3 10 is the only option then, and it still doesn't allow me to do some factoring or cancelling which I believe you have to do.

Answer 14

if you the same as I did before only now the place where the 9 was, its that new expression \[\large \frac {9\log_3 x}{\log_3 10}\]

you should still have x=1 as a solution. but the other factor is just another quadratic...

Answer 15

Could you repost that equation? Seems like you deleted it.

Answer 16

oh i did because it was base 10 and not base 3.... i'll put it up with the new expression...

Answer 17

\[\begin{align}
\log_3(x^3)&=9\log_{10}(x)\\
3\log_3(x)&=9\log_{10}(x)\\
\log_3(x)&=3\log_{10}(x)\\
&=3\frac{\log_3(x)}{\log_3(10)}
\end{align}\]therefore:\[\log_3(10)\log_3(x)=3\log_3(x)\]the only solution to this is to have:\[\log_3(x)=0\]therefore:\[x=3^0=1\]

Answer 18

@asnaseer - Please correct me if I am wrong but, I thought \[\large{(\log_3 x)^3 \neq \log_3 x^3}\]

Answer 19

sorry - misread question - let me try again... :)

Answer 20

\[\begin{align}
(\log_3(x))^3&=9\log_{10}(x)\\
&=9\log_{10}(x)\\
&=3\log_{10}(x)\\
&=3\frac{\log_3(x)}{\log_3(10)}
\end{align}\]let:\[y=\log_3(x)\]therefore:\[y^3=\frac{3y}{\log_3(10)}\]\[y^3-\frac{3y}{\log_3(10)}=0\]\[y(y^2-\frac{3}{\log_3(10)})=0\]therefore:\[y=0\]or:\[y=\pm\sqrt{\frac{3}{\log_3(10)}}\]and you can then get x from here...

Answer 21

sorry - replace the constant 3 above with 9 - copy-paste error

Answer 22

i.e.:\[\begin{align}
(\log_3(x))^3&=9\log_{10}(x)\\
&=9\log_{10}(x)\\
&=9\frac{\log_3(x)}{\log_3(10)}
\end{align}\]which leads to:

y = 0

or:\[y=\pm\sqrt{\frac{9}{\log_3(10)}}=\pm\frac{3}{\sqrt{\log_3(10)}}\]

Answer 23

its quite late at night here which is why I am making silly mistakes :)

Answer 24

you can get x back from:\[y=\log_3(x)\]

Answer 25

so:\[x=3^y\]

Answer 26

Thank you! I'm away from my desk at the moment, but the steps seem to make sense to me.

Answer 27

yw