How to Solve Equations with Unknown Exponents?

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Ok i know this should be a relatively simple equation but i haven't done this stuff in so long i can't remember how to solve it. here goes.

4^(x+2) = 7^x +681

Solve for x

I know the answer is 3 from simple trial and error but i can't show the work to prove it and i know that harder problems exist so i would like to know the procedure. I'm pretty sure that either logs or natural logs (LN) is required somehow. Any help would be greatly appreciated.

Thanks.
 
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well i figure the physics people should have no problem with this.

Ok i know this should be a relatively simple equation but i haven't done this stuff in so long i can't remember how to solve it. here goes.

4^(x+2) = 7^x +681

Solve for x

I know the answer is 3 from simple trial and error but i can't show the work to prove it and i know that harder problems exist so i would like to know the procedure. I'm pretty sure that either logs or natural logs (LN) is required somehow. Any help would be greatly appreciated.

Thanks.
 
there's no easy way to solve an equation that has a polynomial on one side and a transcendental function on the other. Sorry.

P.S. there is another solution near 4.8622 too, but no other solutions besides that one.
 

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