To solve the given equation:
[tex]\[
\log_6(4x^2) - \log_6(x) = 2
\][/tex]
we can use properties of logarithms to simplify and solve for [tex]\( x \)[/tex]. Follow these steps:
1. Apply the properties of logarithms: Use the property that [tex]\( \log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right) \)[/tex].
[tex]\[
\log_6(4x^2) - \log_6(x) = \log_6\left(\frac{4x^2}{x}\right)
\][/tex]
2. Simplify the argument of the logarithm:
[tex]\[
\frac{4x^2}{x} = 4x
\][/tex]
So the equation becomes:
[tex]\[
\log_6(4x) = 2
\][/tex]
3. Convert the logarithmic equation to an exponential equation: Use the property that if [tex]\( \log_b(A) = C \)[/tex], then [tex]\( b^C = A \)[/tex].
[tex]\[
6^2 = 4x
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
36 = 4x
\][/tex]
[tex]\[
x = \frac{36}{4}
\][/tex]
[tex]\[
x = 9
\][/tex]
Hence, the solution to the equation [tex]\(\log_6(4x^2) - \log_6(x) = 2\)[/tex] is:
[tex]\[
\boxed{9}
\][/tex]
