To solve the equation [tex]\( \log_6(4x^2) - \log_6(x) = 2 \)[/tex], we will follow these steps:
1. Use Properties of Logarithms:
The properties of logarithms tell us that [tex]\(\log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right)\)[/tex]. Applying this property, we get:
[tex]\[
\log_6\left(\frac{4x^2}{x}\right) = \log_6(4x) = 2
\][/tex]
2. Rewriting the Equation:
Now, we have a simpler logarithmic equation to work with:
[tex]\[
\log_6(4x) = 2
\][/tex]
3. Convert Logarithmic Form to Exponential Form:
Recall that [tex]\( \log_b(a) = c \)[/tex] implies that [tex]\( b^c = a \)[/tex]. Using this, we convert the logarithmic equation to its exponential form:
[tex]\[
6^2 = 4x
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Calculate [tex]\( 6^2 \)[/tex]:
[tex]\[
36 = 4x
\][/tex]
Now, solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{36}{4} = 9
\][/tex]
So, the solution to the equation [tex]\( \log_6(4x^2) - \log_6(x) = 2 \)[/tex] is [tex]\( x = 9 \)[/tex].
Among the given options:
- [tex]\( x = \frac{1}{12} \)[/tex]
- [tex]\( x = \frac{3}{2} \)[/tex]
- [tex]\( x = 3 \)[/tex]
- [tex]\( x = 9 \)[/tex]
The correct solution is [tex]\( x = 9 \)[/tex].
