If log 3 = A and log 7 = B, findlog_7 9in terms of A and B.

Question that can not be solve analytically .
You can know that the solution of this non-linear simultaneous equations is y=2 and x=3; but the question is : How can mathematically ( algebraically ) find this.
$\begin{array}{lcl}{x}^y& =& 9\\ y^x& =& 8\end{array}$

How do I solve for V in this equation?
${\displaystyle \frac{\ln (T+1)-\ln (V-S+1)}{\ln (V-S+1)}=\frac{1}{K}-1}$
What are the steps if I want to solve for V?

Find intersection of linear and logarithmic lines
I have equations for two lines, one of which is linear and the other is logarithmic, ie:
${\displaystyle y=m_{1}x+c_1}$
${\displaystyle y=m_2\cdot \ln \left(x\right)+c_2}$
..and I need to find out where (if at all) these lines intersect. I realise that I need to solve:
${\displaystyle m_1\cdot x+c_1=m_2\cdot \ln \left(x\right)+c_2}$
..for x, but apart from shuffling the constants around I'm not sure how to do this
Is there a general solution to this problem?
Thanks

Proving Logarithmic identity
Today, I read these two new Logarithmic identity
${\displaystyle a^{{\log }_{a}m}=m}$
${\displaystyle {\log }_{a^q}{m}^p=\frac{p}{q}{\log }_{a}m}$
Both of them seems new to me,so even after solving some problems (directly) based on thesm I haven't fully understood how they holds good,Could anybody show me how to prove them ?

Write the expression as a sum and/or difference of logarithms. Express powers as factors.
${\displaystyle \log \left[\frac{x(x+6)}{{(x+8)}^4}\right],x>0}$

If ${\displaystyle f\left(x\right)={\log }_{a}x}$, show that ${\displaystyle \frac{f(x+h)-f\left(x\right)}{h}={\log }_a{(1+\frac{h}{x})}^{\frac{1}{h}},h\ne c0}$