Can you please compare the numbers 3^log4 of 5 and 5^ log4 of 3, just to make it clear the bases are equal and are 4. Please help me with this problem
I used a circuitous method involving taking the log of both sides and then using the change of base rule.
$\displaystyle 3^{\log_4(5)} = 5^{\log_4(3)}$
$\displaystyle \log_4(5)\ln(3) = \log_4(3)\ln(5)$
$\displaystyle \dfrac{\ln(5)\ln(3)}{\ln(4)} = \dfrac{\ln(3)\ln(5)}{\ln(4)}$
Hence they're equal.
Hello, Jkanariya!
$\displaystyle \text{Compare }\:3^{\log_45}\,\text{ and }\,5^{\log_43}$
They are equal!
$\displaystyle \text{Let }\,3^{\log_45} \,=\,P$
$\displaystyle \text{Take logs, base 4:}$
. . $\displaystyle \log_4\left(3^{\log_45}\right) \:=\:\log_4P \quad\Rightarrow\quad (\log_45)(\log_43) \:=\:\log_4P \;\;{\bf[1]} $
$\displaystyle \text{Let }\,5^{\log_43} \,=\,Q$
$\displaystyle \text{Take logs, base 4:}$
. . $\displaystyle \log_4\left(5^{\log_43}\right) \:=\:\log_4Q \quad\Rightarrow\quad (\log_43)(\log_45) \:=\:\log_4Q\;\;{\bf[2]}$
$\displaystyle \text{Equate }{\bf[1]}\text{ and }{\bf[2]}\!:\;\;\log_4P \:=\:\log_4Q \quad\Rightarrow\quad P \:=\:Q$
$\displaystyle \text{Therefore: }\:3^{\log_45} \;=\;5^{\log_43}$