Mastering IGCSE Mathematics: Solving Exponential Equations | Solve for x in 5^x=1

Unlock the secrets of exponential equations in IGCSE Mathematics with this comprehensive tutorial. Join us as we break down the process of solving equations like 5^x=1, providing clear, step-by-step instructions and insights. Strengthen your math skills and ace your exams with this in-depth guide. Subscribe now for more IGCSE math tutorials and boost your confidence in tackling challenging problems!
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[Skank_and_Gutterboy]

You can solve this in half a second if you remember that anything to the zero-power
equals 1. Solving it the long(er) way, take the log of both sides:

x log(5) = log(1)
x = log(1)/log(5)
log(1) = 0, so x = 0.

You have to worry a little bit that there may be more than one solution. Exponential functions sometimes get mysterious between 0 and 1 and finding intersection(s) of two exponential functions can get complicated and have multiple solutions. With any constant to the x-power and you want to find where it equals a constant, you won't have that problem. When x is decreasing, it goes asymptotic with the x-axis. When x is increasing, it keeps going up and up and it's not turning back toward the x-axis at any point (you can prove all this with derivatives). So, the function y=5^x crosses y=1 only once. x=0, you're done.



If you're pressed for time on a test and don't have time to ponder all these nuances, take the easy solution x=0 and move on, next question. Getting half-credit on a question is better than not finishing the test.

[antonnym]

Any non-zero quantity raised to the power of 0 is defined to be 1; therefore x=0 is a solution to this problem.

[jakemccoy]

A more general solution would be nice.

[MrMousley]

ANYTHING 'to the power 0' = 1 So x = 0