Here's the question you clicked on:
geerky42
Find the value of x and y:\[ (2x)^{\ln 2} = (3y)^{\ln 3} \]\[ 3^{\ln x} = 2^{\ln y}\]
Is this a system of equations?
.... @Goten77 hard to read it... what are you doing?
think bout it..... not that hard to read..... but it wont ever end....
\[ \large \begin{array}{rcl} (2x)^{\ln2}&=&(3y)^{\ln3} \\ \ln(2x)\ln(2)&=&\ln(3y)\ln(3) \\ \ln(2)\ln(2) +\ln(2) \ln(x) &=&\ln(3)\ln(y)+\ln(3)\ln(3) \\ \ln(2) \ln(x) - \ln(3)\ln(y) &=&\ln(3)\ln(3)-\ln(2)\ln(2) \end{array} \] \[ \large \begin{array}{rcl} 3^{\ln x}&=&2^{\ln y} \\ \ln(3)\ln (x)&=&\ln(2)\ln (y) \\ \ln(3)\ln (x) - \ln(2)\ln (y) &=& 0 \end{array} \] \[ \begin{bmatrix} \ln(2)&-\ln(3) \\ \ln(3)&- \ln(2) \end{bmatrix} \begin{bmatrix} \ln(x) \\ \ln(y) \end{bmatrix} = \begin{bmatrix} \ln(3)\ln(3)-\ln(2)\ln(2) \\ 0 \end{bmatrix} \]
If you put it into a matrix, it's not as big a mess.
then at the very end raise it to the power of \(e\). It could be a singular matrix though?
strange its basically a property..... memorable