Solving an equation of a sum of terms with the same exponent

Asked Sep 14 '23 at 07:47

Active Sep 14 '23 at 11:00

Viewed 63 times

I have an equation that simplifies to the equation below, that I need to solve for t.

$$ ab^t + cd^t = f $$

All letters represent simple variables.

I don't remember/know enough of the exponentiation rules to figure this out. Neither wolfram alpha (free version) or symbolab.com can figure it out.

Someone asked for the original equation:

$$ \frac{(1+r_1)^t}{r_1}P_1+{P_v}_1(1+r_1)^t=\frac{P_2}{r_2}-(1+r_2)^t(\frac{P_2}{r_2}-{P_v}_2)+o $$ For my uses:
${P_v}_1, {P_v}_2 > 10,000$

$P_1, P_2 > 100$

$0 < r_1,r_2 < 0.1$

edited Sep 14 '23 at 11:00

asked Sep 14 '23 at 07:47

Francisco Canedo

1

You will need numerical methods. If Wolfram Alpha fails , probably even Lambert-W won't help. – Peter Sep 14 '23 at 07:49
Are $(a,b,c,e)$ all positive and greater than $1$? – Claude Leibovici Sep 14 '23 at 08:20
Maybe if you share the original equation we can see other possibilities – César VB Sep 14 '23 at 08:37
$b^t$ and $d^t$ are not generally algebraically dependent (eg, take $(b,d)=(2,3)$). Hence, their sum generally lacks an elementary inverse. – Jam Sep 14 '23 at 08:46
The original equation added by popular demand. – Francisco Canedo Sep 14 '23 at 11:01
As $r<<1$ we can approximate $(1+r)^t \approx 1+ r t+\frac 12r^2(t-1)t$ and then solve a second order polynomial to have a reasonable guess, and then proceed to refine the results. – Cesareo Sep 14 '23 at 11:40

0 Answers0