1. ## logarithims

i'm getting stuck ....a little help please

$\displaystyle 5^{x-2} = 4^x$

i got..
$\displaystyle log 5 ^{x-2} = log 4^x$
$\displaystyle (x-2) log 5 = x log 4$

and i'm stuck here...

2. Originally Posted by extraordinarymachine
i'm getting stuck ....a little help please

$\displaystyle 5^x-2 = 4^x$

i got..
$\displaystyle log 5 ^x-2 = log 4^x$ <<<<<<<<< never commit such a crime again!
$\displaystyle (x-2) log 5 = x log 4$

and i'm stuck here...
Do you mean

$\displaystyle 5^x - 2 = 4^x$ If so

$\displaystyle \left(\dfrac54 \right)^x = 2~\implies~x=\log_{\frac54}(2)=\dfrac{\ln(2)}{\ln( 5)-\ln(4)}$

or

$\displaystyle 5^{x-2} = 4^x$ If so

$\displaystyle \frac1{25} \cdot 5^x = 4^x~\implies~\left(\dfrac54 \right)^x = 25 ~\implies~x=\log_{\frac54}(25)=\dfrac{\ln(25)}{\ln (5)-\ln(4)}$

3. Originally Posted by earboth
Do you mean

$\displaystyle 5^x - 2 = 4^x$ If so

sorry i messed up the question, i ment to write it like this

4. Originally Posted by extraordinarymachine
i'm getting stuck ....a little help please

$\displaystyle 5^{x-2} = 4^x$

i got..
$\displaystyle log 5 ^{x-2} = log 4^x$
$\displaystyle (x-2) log 5 = x log 4$

and i'm stuck here...
$\displaystyle (x-2) log 5 = x log 4 \Leftrightarrow \frac{ x-2}{x }$$\displaystyle = \frac{log4 }{log5 } \Leftrightarrow 1- \frac{ 2}{x } =\frac{log4 }{log5 }$

5. $\displaystyle (x-2)log5=xlog4$

$\displaystyle \frac{x-2}{x}=\frac{log4}{log2}$

$\displaystyle 1-\frac{2}{x}=\frac{log4}{log5}$

$\displaystyle \frac{2}{x}=1-\frac{log4}{log5}$

$\displaystyle \frac{2}{x}=\frac{log5}{log5}-\frac{log4}{log5}$

$\displaystyle \frac{2}{x}=\frac{log5-log4}{log5}$

$\displaystyle \frac{x}{2}=\frac{log5}{log5-log4}$

$\displaystyle x=\frac{2log5}{log5-log4}$

6. 5 steps
$\displaystyle (x-2)log5=xlog4$
$\displaystyle xlog5-2log5=xlog4$
$\displaystyle xlog5-xlog4=2log5$
$\displaystyle x(log5-log4)=2log5$
$\displaystyle x=\frac{2log5}{log5-log4}$

7. Originally Posted by Krahl
5 steps
$\displaystyle (x-2)log5=xlog4$
$\displaystyle xlog5-2log5=xlog4$
$\displaystyle xlog5-xlog4=2log5$
$\displaystyle x(log5-log4)=2log5$
$\displaystyle x=\frac{2log5}{log5-log4}$
this was really helpful thanks!