Answer :
To simplify the expression [tex]\(\log \left(x^3\right) - \log \left(x^2\right)\)[/tex], let's break it down step by step using the properties of logarithms:
Step 1: Apply the Logarithm Subtraction Rule
The logarithm subtraction rule states:
[tex]\[ \log_a(b) - \log_a(c) = \log_a\left(\frac{b}{c}\right) \][/tex]
By using this rule, we can rewrite [tex]\(\log \left(x^3\right) - \log \left(x^2\right)\)[/tex] as:
[tex]\[ \log\left(\frac{x^3}{x^2}\right) \][/tex]
Step 2: Simplify the Argument of the Logarithmic Function
Next, simplify [tex]\(\frac{x^3}{x^2}\)[/tex]:
[tex]\[ \frac{x^3}{x^2} = x^{3-2} = x^1 = x \][/tex]
Step 3: Write the Resulting Logarithmic Expression
Now we substitute back into the logarithmic expression:
[tex]\[ \log\left(\frac{x^3}{x^2}\right) = \log(x) \][/tex]
Thus, the simplified form of [tex]\(\log \left(x^3\right) - \log \left(x^2\right)\)[/tex] is:
[tex]\[ \log(x) \][/tex]
So the answer is:
[tex]\[ \boxed{\text{D. } \log(x)} \][/tex]
Step 1: Apply the Logarithm Subtraction Rule
The logarithm subtraction rule states:
[tex]\[ \log_a(b) - \log_a(c) = \log_a\left(\frac{b}{c}\right) \][/tex]
By using this rule, we can rewrite [tex]\(\log \left(x^3\right) - \log \left(x^2\right)\)[/tex] as:
[tex]\[ \log\left(\frac{x^3}{x^2}\right) \][/tex]
Step 2: Simplify the Argument of the Logarithmic Function
Next, simplify [tex]\(\frac{x^3}{x^2}\)[/tex]:
[tex]\[ \frac{x^3}{x^2} = x^{3-2} = x^1 = x \][/tex]
Step 3: Write the Resulting Logarithmic Expression
Now we substitute back into the logarithmic expression:
[tex]\[ \log\left(\frac{x^3}{x^2}\right) = \log(x) \][/tex]
Thus, the simplified form of [tex]\(\log \left(x^3\right) - \log \left(x^2\right)\)[/tex] is:
[tex]\[ \log(x) \][/tex]
So the answer is:
[tex]\[ \boxed{\text{D. } \log(x)} \][/tex]
