How to solve equations with two logarithmic terms?

Question

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.

I mean the logs are to the power Q and Z respectively, outside yes — Wiebe Kuper, May 02 '18 at 13:02
Your issue is not solving with a logarithm. Just state that $u=\log t$ and see what is going on. — Martigan, May 02 '18 at 13:05
Can you give some more details about $a,b$. Are these integers? Are they coprimes? — Martigan, May 02 '18 at 13:14

score 1 · Accepted Answer · answered May 02 '18 at 17:54

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 $\alpha, \beta, \gamma$

1 Answers1