Let's solve the equation [tex]\( 4^x = \left( \frac{1}{8} \right)^{x+5} \)[/tex].
First, rewrite both sides with a common base.
Notice that:
- [tex]\(4\)[/tex] can be expressed as [tex]\(2^2\)[/tex],
- [tex]\(\frac{1}{8}\)[/tex] can be expressed as [tex]\( \left( \frac{1}{2^3} \right) = 2^{-3} \)[/tex].
Thus, rewrite the equation as:
[tex]\[
4^x = \left( \frac{1}{8} \right)^{x+5}
\][/tex]
which becomes:
[tex]\[
(2^2)^x = (2^{-3})^{x+5}
\][/tex]
Using the property of exponents [tex]\((a^m)^n = a^{mn}\)[/tex], rewrite the equation:
[tex]\[
2^{2x} = 2^{-3(x+5)}
\][/tex]
Since the bases are the same, we can equate the exponents:
[tex]\[
2x = -3(x + 5)
\][/tex]
Now solve for [tex]\(x\)[/tex]:
[tex]\[
2x = -3x - 15
\][/tex]
Add [tex]\(3x\)[/tex] to both sides:
[tex]\[
2x + 3x = -15
\][/tex]
Combine like terms:
[tex]\[
5x = -15
\][/tex]
Divide both sides by 5:
[tex]\[
x = -3
\][/tex]
After solving the equation, we've determined that the real solution to [tex]\( 4^x = \left( \frac{1}{8} \right)^{x+5} \)[/tex] is [tex]\( x = -3 \)[/tex].
So the value of [tex]\( x \)[/tex] is:
[tex]\[
\boxed{-3}
\][/tex]
Additionally, according to the provided numerical results, there are complex solutions as well:
- [tex]\(x = -3\)[/tex]
- [tex]\(x = -3.0 - 3.62588811346175i\)[/tex]
- [tex]\(x = -3.0 - 1.81294405673088i\)[/tex]
- [tex]\(x = -3.0 + 1.81294405673088i\)[/tex]
- [tex]\(x = -3.0 + 3.62588811346175i\)[/tex]
Among the given multiple-choice answers, the answer is:
[tex]\[
\boxed{-3}
\][/tex]
