Certainly! Let's expand the logarithm [tex]\(\log (4 x^2)\)[/tex] fully using the properties of logarithms, and express the final answer in terms of [tex]\(\log x\)[/tex].
### Step-by-step Solution:
1. Given Logarithmic Expression:
[tex]\[
\log (4 x^2)
\][/tex]
2. Separate the Product Inside the Logarithm:
Using the logarithmic property [tex]\(\log(ab) = \log a + \log b\)[/tex], we can separate the product inside the logarithm:
[tex]\[
\log (4 x^2) = \log 4 + \log (x^2)
\][/tex]
3. Bring the Exponent Outside:
Using the logarithmic property [tex]\(\log (a^b) = b \log a\)[/tex], we can bring the exponent of [tex]\(x^2\)[/tex] outside the logarithm:
[tex]\[
\log (x^2) = 2 \log x
\][/tex]
4. Substitute Back into the Expression:
Now substituting [tex]\(\log (x^2) = 2 \log x\)[/tex] back into the expression, we get:
[tex]\[
\log 4 + \log (x^2) = \log 4 + 2 \log x
\][/tex]
### Final Answer:
[tex]\[
\log (4 x^2) = \log 4 + 2 \log x
\][/tex]
So, the logarithm [tex]\(\log (4 x^2)\)[/tex] is fully expanded as [tex]\(\log 4 + 2 \log x\)[/tex] using the properties of logarithms.
