Solve the simultaneous equations, giving your answer in exact form.

$2\ln x+\ln y=1+\ln 5$

$\ln 10x-\ln y=2+\ln 2$

Solve the simultaneous equations, giving your answer in exact form.

$e^{3x+4y}=2e^{2x-y}$

$e^{2x+y}=8e^{x+6y}$

Solve $\ln (2x+1)\le \ln (x+4)$.

Given that $a$ and $b$ are constants, use logarithms to change each of these non-linear equations into the form $Y=mX+c$. State what the variables $X$ and $Y$ and the constants $m$ and $c$ represent. (Note: there might be more than one method to do this.)

$y={\mathrm{e}}^{ax+b}$

Given that $a$ and $b$ are constants, use logarithms to change each of these non-linear equations into the form $Y=mX+c$. State what the variables $X$ and $Y$ and the constants $m$ and $c$ represent. (Note: there might be more than one method to do this.)

$y={10}^{ax-b}$

Given that $a$ and $b$ are constants, use logarithms to change each of these non-linear equations into the form $Y=mX+c$. State what the variables $X$ and $Y$ and the constants $m$ and $c$ represent. (Note: there might be more than one method to do this.)

$y=a\left(x^{-b}\right)$

Given that $a$ and $b$ are constants, use logarithms to change each of these non-linear equations into the form $Y=mX+c$. State what the variables $X$ and $Y$ and the constants $m$ and $c$ represent. (Note: there might be more than one method to do this.)

$y=a\left(b^x\right)$

Given that $a$ and $b$ are constants, use logarithms to change each of these non-linear equations into the form $Y=mX+c$. State what the variables $X$ and $Y$ and the constants $m$ and $c$ represent. (Note: there might be more than one method to do this.)

$a={\mathrm{e}}^{x^2+by}$

Given that $a$ and $b$ are constants, use logarithms to change each of these non-linear equations into the form $Y=mX+c$. State what the variables $X$ and $Y$ and the constants $m$ and $c$ represent. (Note: there might be more than one method to do this.)

$x^a{y}^b=8$