Once again I return with questions about logarithms. This time I am having trouble with solving equations of the following form:
$a\cdot \log(t)^{Q} - b\cdot \log(t)^{Z} = R $
I cannot figure out how to solve this equation for $t$. What I do know is the following: taking the exponential on both sides results in
$\exp(a\cdot \log(t)^{Q}) = \exp(R+ b\cdot \log(t)^{Z}) $ $\iff$ $\exp(a\cdot \log(t)^{Q}) = e^{R}\cdot e^{ b\cdot \log(t)^{Z}}.$
Thanks in advance.
The "most" you can do is to define $x = \log(t)^Q$, then your equation is
$x = \frac{b}{a} x^{\frac{Z}{Q}} + \frac{R}{a}$
That's it, you want the solution to
$x = \alpha x^{\beta} + \gamma$
Sadly, the solution to this equation cannot be written in term of usual functions.
But you can calculate its approximate value for specific values of $\alpha, \beta, \gamma$