Sure, let's solve the equation [tex]\(16^x = \frac{1}{\sqrt{x}}\)[/tex] step-by-step.
1. Rewrite the equation in terms of exponents and simpler forms:
[tex]\[
16^x = \frac{1}{\sqrt{x}}
\][/tex]
We know that [tex]\(16 = 2^4\)[/tex], so we can replace 16 with [tex]\(2^4\)[/tex]:
[tex]\[
(2^4)^x = x^{-\frac{1}{2}}
\][/tex]
Simplifying the left-hand side:
[tex]\[
2^{4x} = x^{-\frac{1}{2}}
\][/tex]
2. Apply the natural logarithm to both sides to simplify:
[tex]\[
\ln(2^{4x}) = \ln(x^{-\frac{1}{2}})
\][/tex]
Using the properties of logarithms:
[tex]\[
4x \ln(2) = -\frac{1}{2} \ln(x)
\][/tex]
3. Isolate the logarithm involving [tex]\(x\)[/tex]:
[tex]\[
4x \ln(2) = -\frac{1}{2} \ln(x)
\][/tex]
Multiply both sides by -2 to simplify:
[tex]\[
-8x \ln(2) = \ln(x)
\][/tex]
4. Rewrite the equation in exponential form:
[tex]\[
e^{\ln(x)} = e^{-8x \ln(2)}
\][/tex]
Since [tex]\(e^{\ln(x)} = x\)[/tex], we have:
[tex]\[
x = e^{-8x \ln(2)}
\][/tex]
5. Solve for [tex]\(x\)[/tex] numerically:
Solving this transcendental equation exactly requires numerical methods or special functions that can be complex. However, through numerical analysis, the valid solution for [tex]\(x\)[/tex] is:
[tex]\[
x = 0.25
\][/tex]
So, the solution to the equation [tex]\(16^x = \frac{1}{\sqrt{x}}\)[/tex] is:
[tex]\[
x = 0.25
\][/tex]
