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Solve the equation: 5^(x-3)=3^(x+2)?
1 Answer
- MathMan TGLv 710 years agoFavorite Answer
Since 5 and 3 are primes and therefore relatively prime,
the value for x won't be an integer.
So we use logs or ln's
(x - 3) ln 5 = (x + 2) ln 3
(x - 3) / (x + 2) = ln 3 / ln 5 = 0.6826
x - 3 = 0.6826 (x + 2)
x - 3 = 0.6826 x + 1.3652
.3174 x = 4.3652
x = 13.753
(all numbers rounded)
5 ^ (10.753) = 32 811 377
3 ^ (15.753) = 32 816 414.3
Difference due to round off errors.
Keeping it more exact:
x - 3 = (ln 3) / (ln 5) (x + 2)
x - 3 = (ln 3) / (ln 5) x + 2 ln 3 / ln 5
x ( 1 - ln 3 / ln 5) = 3 + 2 ln 3 / ln 5
x = (3 + 2 ln 3 / ln 5) / ( 1 - ln 3 / ln 5)
x = 13.7533005
Now 5^(x-3) = 32 827 249.6
and 3^(x+2) = 32 827 249.8
A little more accuracy goes a long way.