How do you solve 4^(x-7)+7=43?

Aug 8, 2016

$x = 9.5850$

Explanation:

If the answer is one of the exact powers, we can calculate the value of x. If it is not, and the variable is in the index, we need to use logs.

${4}^{x - 7} = 36$

Although 4 is 2^2 and 36 is 6^2, the bases are different, use logs:

${2}^{2 \left(x - 7\right)} = {6}^{2}$

$2 \left(x - 7\right) \log 2 = 2 \log 6$

$\left(x - 7\right) \log 2 = \frac{2 \log 6}{2}$

$x - 7 = \frac{\log 6}{\log 2} = 2.5850$

$x = 9.5850$

Note: we could also just have used $\frac{\log 36}{\log 4}$ and got to the same answer, but it is always worth exploring the numbers a bit, because often a simple solution reveals itself.