How to Solve Equations with Unknown Exponents?

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SUMMARY

The equation 4^(x+2) = 7^x + 681 can be solved using logarithmic methods. The confirmed solution is x = 3, with another approximate solution at x ≈ 4.8622. The discussion emphasizes that equations combining polynomial and transcendental functions do not have straightforward solutions, highlighting the necessity of logarithms in the solving process.

PREREQUISITES
  • Understanding of exponential equations
  • Familiarity with logarithmic functions, specifically natural logarithms (LN)
  • Basic algebraic manipulation skills
  • Knowledge of polynomial and transcendental functions
NEXT STEPS
  • Study the properties of logarithms and their applications in solving equations
  • Learn techniques for solving exponential equations
  • Explore numerical methods for approximating solutions to complex equations
  • Investigate the differences between polynomial and transcendental functions
USEFUL FOR

Students, educators, and anyone interested in advanced algebraic techniques, particularly those dealing with exponential equations and logarithmic solutions.

foxyman20
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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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