Solve exponential equation for x

Question

I am stuck on solving

$1000^x - 2 * 100^x = 3 * 10^x$

for x. I am sure I learned how to do that, but it is goooone. I have a result, symbolic and numeric, using Wolfram Alpha, but the step-by-step solution is not only available to pro users.

I understand I need to bring the equation onto the same base, as suggest as first step by Wolfram Alpha. Something like

$e^{ln(1000^x)}-e^{ln(2)}*e^{ln(100^x)}=e^{ln(10^x)}$

which can be simplified to

$e^{x*ln(1000)}-e^{ln(2)}*e^{x*ln(100)}=e^{x*ln(10)}$

But I am stuck there. I would "drop" (logarithmicate?) the whole equation at some point to get x out of the exponents, and then solve it easily - but I do not know how to do that as long as there is this sum on the left side.

I also seem to have forgotten if (and how) the terms with $e^{x*ln(...)}$ can be split in two.

Any pointers? Thanks a lot!

Answer 1

Set $10^x = t$. Then we have:

$$t^3 - 2t^2-3t = 0$$

Also, keep in mind that $t > 0$, so

$$t^2-2t-3=0$$

Answer 2

Hint:

write your equation as $$ 10^{3x}-2\cdot 10^{2x}-3\cdot 10^x=0 \iff 10^x(10^{2x}-2\cdot10^x-3)=0 $$

and solve for $y=10^x \ne 0$