## geerky42 2 years ago Find the value of x and y:$(2x)^{\ln 2} = (3y)^{\ln 3}$$3^{\ln x} = 2^{\ln y}$

1. Goten77

hmm

2. Goten77

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3. wio

Is this a system of equations?

4. Goten77

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5. Goten77

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6. wio

.... @Goten77 hard to read it... what are you doing?

7. Goten77

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8. Goten77

think bout it..... not that hard to read..... but it wont ever end....

9. wio

$\large \begin{array}{rcl} (2x)^{\ln2}&=&(3y)^{\ln3} \\ \ln(2x)\ln(2)&=&\ln(3y)\ln(3) \\ \ln(2)\ln(2) +\ln(2) \ln(x) &=&\ln(3)\ln(y)+\ln(3)\ln(3) \\ \ln(2) \ln(x) - \ln(3)\ln(y) &=&\ln(3)\ln(3)-\ln(2)\ln(2) \end{array}$ $\large \begin{array}{rcl} 3^{\ln x}&=&2^{\ln y} \\ \ln(3)\ln (x)&=&\ln(2)\ln (y) \\ \ln(3)\ln (x) - \ln(2)\ln (y) &=& 0 \end{array}$ $\begin{bmatrix} \ln(2)&-\ln(3) \\ \ln(3)&- \ln(2) \end{bmatrix} \begin{bmatrix} \ln(x) \\ \ln(y) \end{bmatrix} = \begin{bmatrix} \ln(3)\ln(3)-\ln(2)\ln(2) \\ 0 \end{bmatrix}$

10. wio

If you put it into a matrix, it's not as big a mess.

11. wio

then at the very end raise it to the power of $$e$$. It could be a singular matrix though?

12. Goten77

strange its basically a property..... memorable