How do I isolate $k$ in the following equation?

Question

I tried to isolate the unknown $k$ in the the following equation by using logarithms, but my resolution was ugly. I am trying to learn mathematics by my own (again!), because it's a beautiful subject and I used to be good at it.

This week I've gotten back to my studies in math.

I tried solving the equation $r = q ^{(i/j)^k}$ for k, but I don't know where to go from the step 4:

Step 1

$\ln(r) = (i/j)^k\ln(q)$

Step 2

$(i/j)^k = \ln(r)/\ln(q)$

Step 3

I took the logarithms again(!).

$\ln((i/j)^k) = \ln(\ln(r)/\ln(q))$

Step 4

$k*\ln(i/j) = \ln(\ln(r)/\ln(q))$

Is there any property that can help me to isolate $k$ elegantly?

Thank you in advance!

I don't know if this is allowed, but THANK YOU! This is so motivating! You guys are awesome!

Answer 1

The solution you have found is correct, and seems to be the most 'elegant' one, though 'elegance' is of course subject to debate. You could write $$k=\frac{\ln(\ln(r)/\ln(q))}{\ln(i/j)}=\frac{\ln\ln r-\ln\ln q}{\ln i-\ln j}.$$