# How do you solve 8^(2x) = 8^(x+7)?

Apr 5, 2016

$x = 7$

#### Explanation:

Note that since the bases of both exponential functions are equal, their exponents must also be equal.

This gives us, once we set their exponents equal to one another:

$2 x = x + 7 \text{ "=>" } x = 7$

If we want to formalize this, take the logarithm with base $8$ of both sides to undo the exponential functions.

${\log}_{8} \left({8}^{2 x}\right) = {\log}_{8} \left({8}^{x + 7}\right) \text{ "=>" } 2 x = x + 7$

Which is the equation we saw previously.

We can generalize this idea of exponential functions having the same bases as saying that if:

${a}^{b} = {a}^{c} \text{ "=>" } b = c$