To solve for [tex]\( x \)[/tex] in the given logarithmic equation [tex]\( \log_x 8 = 0.5 \)[/tex], we can follow these steps:
1. Understand the definition of the logarithm:
The expression [tex]\( \log_x 8 = 0.5 \)[/tex] states that the logarithm of 8 with base [tex]\( x \)[/tex] is 0.5. This means that [tex]\( x \)[/tex] raised to the power of 0.5 equals 8.
2. Rewrite the logarithmic equation as an exponential equation:
By the definition of logarithms, [tex]\( \log_x 8 = 0.5 \)[/tex] is equivalent to:
[tex]\[
x^{0.5} = 8
\][/tex]
3. Eliminate the fractional exponent:
To solve for [tex]\( x \)[/tex], we need to get rid of the exponent 0.5. We do this by squaring both sides of the equation. Squaring both sides yields:
[tex]\[
\left( x^{0.5} \right)^2 = 8^2
\][/tex]
[tex]\[
x = 64
\][/tex]
4. Verify the solution:
Substitute [tex]\( x = 64 \)[/tex] back into the original logarithmic equation to ensure it satisfies the equation:
[tex]\[
\log_{64} 8 = 0.5
\][/tex]
We can check this by converting to exponential form:
[tex]\[
64^{0.5} = 8
\][/tex]
[tex]\[
8 = 8
\][/tex]
This confirms that our solution is correct.
Therefore, the value of [tex]\( x \)[/tex] is 64, which corresponds to option D.
