To solve the equation:
[tex]\[3^{x^2 - 8x + 15} = 81^{2 - 4x}\][/tex]
we need to use some properties of exponents and logarithms for solving it step by step.
1. Notice that [tex]\(81 = 3^4\)[/tex]. Therefore, we can rewrite [tex]\(81^{2 - 4x}\)[/tex] as:
[tex]\[
81^{2 - 4x} = (3^4)^{2 - 4x} = 3^{4(2 - 4x)} = 3^{8 - 16x}
\][/tex]
2. Now, we have the equation:
[tex]\[
3^{x^2 - 8x + 15} = 3^{8 - 16x}
\][/tex]
Because the bases (3) are the same, we can equate the exponents:
[tex]\[
x^2 - 8x + 15 = 8 - 16x
\][/tex]
3. Simplify the equation:
[tex]\[
x^2 - 8x + 15 = 8 - 16x
\][/tex]
Move all terms to one side to set the equation to zero:
[tex]\[
x^2 - 8x + 15 - 8 + 16x = 0
\][/tex]
Combine like terms:
[tex]\[
x^2 + 8x + 7 = 0
\][/tex]
4. Factor the quadratic equation:
[tex]\[
x^2 + 8x + 7 = (x + 7)(x + 1) = 0
\][/tex]
5. Set each factor equal to zero:
[tex]\[
x + 7 = 0 \quad \text{or} \quad x + 1 = 0
\][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[
x = -7 \quad \text{or} \quad x = -1
\][/tex]
Thus, the solutions to the equation are:
[tex]\[
x = -7, -1
\][/tex]
