We are given the exponential equation shown below.
8^(x-4)=1
Recall that can solve an exponential equation with unlike bases by rewriting it using powers with a common base. We can use the Zero Exponent Definition to achieve this. This property states that any nonzero number raised to the zeroth power is equal to one.
Zero Exponent Definition
a^0 =1
In particular we can use this to rewrite 1 as 8^0.
8^(x-4)=1 ⇔ 8^(x-4)=8^0
According to the Property of Equality for Exponential Equations , two powers with the same base are equal if and only if their exponents are equal as well. Therefore, we know that the exponents must be equal.
8^(x-4)=8^0 ⇔ x-4=0
From the equation found above, we can conclude that the solution for our original exponential equation is x =4.
