How do you solve #8^(2x) = 8^(x+7)#?
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:
#2x=x+7" "=>" "x=7#
If we want to formalize this, take the logarithm with base
#log_8(8^(2x))=log_8(8^(x+7))" "=>" "2x=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" "=>" "b=c#