Solving equation with logs

• Sep 12th 2010, 10:04 AM
metaname90
Solving equation with logs
Attachment 18904
With the help of technology I've figured out that r=27 but I can't quite work it out.

I know 2^3log4(3)=2^log4(27 or 2^(log27)/(log4) and sqrt(r)=r^(1/2) but I can't quite put it all together....
• Sep 12th 2010, 10:23 AM
Plato
First notice that $\displaystyle \log_4(2)=\frac{1}{2}$
Then $\displaystyle \displaystyle 2^{3\log_4(3)}= 2^{\log_4(27)}$.
Take the log of both $\displaystyle \log_4(27)\log_4(2)= \frac{1}{2}\log_4(27)= \log_4(\sqrt{27})=\log_4(\sqrt{r})$
• Sep 12th 2010, 01:30 PM
Soroban
Hello, metaname90!

Quote:

$\displaystyle \text{Solve for }r\!:\;\;2^{3\log_4(3)} \:=\:\sqrt{r}$

The left side is: .$\displaystyle 2^{\log_4(3^3)} \;=\;2^{\log_4(27)}$

Since $\displaystyle 2 = 4^{\frac{1}{2}}$
. . we have: .$\displaystyle (4^{\frac{1}{2}})^{\log_4(27)} \;=\; 4^{\frac{1}{2}\log_4(27)} \;=\;4^{\log_4(27^{\frac{1}{2}})} \;=\;27^{\frac{1}{2}}$

The equation becomes: .$\displaystyle 27^{\frac{1}{2}} \:=\:r^{\frac{1}{2}} \quad\Rightarrow\quad r \:=\:27$