Exponential Variables

Question

7^x=6^(x+7)

Solve for x.

Answer 1

7^x = 6^(x+7)

log (7^x) = log (6^(x+7))

x log 7 = (x+7)log 6

x log 7 = xlog6 + 7log6

x log 7 - xlog6 = 7 log6

x(log 7 - log 6) = 7 log 6

x = 7 log 6/(log7 - log6)

x = 7 log 6(log 7/6)

Answer 2

We start by taking a log of each side. It actually doesn't matter what log base you use, as long as you follow the logarithm rules, so I'll pick the natural log:

7^x = 6^x+7

⇒ ln ( 7^x) = ln (6^x+7)

⇒ x * ln 7 = (x+7) * ln 6 (by the power property of logarithms).

Now it's just a little algebra. We collect all the "x" on the left side:

x * ln 7 = (x+7) * ln 6

⇒ x * ln 7 = x*ln 6 + 7*ln 6

⇒ (ln7 - ln6) x = 7*ln 6

⇒ x = 7 * ln6 / (ln7 - ln 6)

= 7 * ln6 / ln(7/6).

And we're done!