To solve the logarithmic equation [tex]\(2 \log x = \log 64\)[/tex], let's follow a step-by-step approach:
1. Rewrite the given equation using properties of logarithms:
The equation is:
[tex]\[
2 \log x = \log 64
\][/tex]
2. Apply the power rule of logarithms:
Using the power rule of logarithms, [tex]\(a \log b = \log b^a\)[/tex], we can rewrite the left side of the equation:
[tex]\[
\log x^2 = \log 64
\][/tex]
3. Equate the arguments of the logarithms:
Since the logarithms are equal, their arguments must be equal as well:
[tex]\[
x^2 = 64
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
To find [tex]\(x\)[/tex], we take the square root of both sides:
[tex]\[
x = \sqrt{64} \quad \text{or} \quad x = -\sqrt{64}
\][/tex]
Simplifying, we get:
[tex]\[
x = 8 \quad \text{or} \quad x = -8
\][/tex]
5. Consider the domain of the logarithmic function:
Since [tex]\(x\)[/tex] appears inside a logarithm, it must be positive ([tex]\(x > 0\)[/tex]). Therefore, [tex]\(x = -8\)[/tex] is not a valid solution in this context.
The only valid solution is:
[tex]\[
x = 8
\][/tex]
6. Verify with the given multiple choices:
Comparing our solution with the given choices:
[tex]\[
x = 1.8, \quad x = 8, \quad x = 32, \quad x = 128
\][/tex]
We find that the correct choice from the given options is:
[tex]\[
x = 8
\][/tex]
In conclusion, after following the steps to solve the equation [tex]\(2 \log x = \log 64\)[/tex], we find that the valid solution is:
[tex]\[
\boxed{8}
\][/tex]
