Solving Exponential Equations with Same Base
To solve the equation \( 8^{x+1} = 2^{5-x} \) using a common base, we first need to express both sides of the equation with the same base. We know that 8 is equivalent to \( 2^3 \). Thus, we can rewrite \( 8^{x+1} \) as \( (2^3)^{x+1} \), which simplifies to \( 2^{3(x+1)} \).
Now the equation becomes:
\( 2^{3(x+1)} = 2^{5-x} \)
Since the bases are now the same, we can set the exponents equal to each other:
\( 3(x+1) = 5 - x \)
Next, we distribute the 3 on the left side:
\( 3x + 3 = 5 - x \)
Now bring all x terms to one side and constants to the other side:
\( 3x + x = 5 - 3 \)
Combine like terms:
\( 4x = 2 \)
Finally, divide by 4 to get x alone:
\( x = \frac{2}{4} \)
Simplify the fraction:
\( x = \frac{1}{2} \)
So the solution is x = 1/2.
