Simplify the following expression:

[tex]\(\log \left(x^3\right)-\log \left(x^2\right)\)[/tex]

A. [tex]\(\log \left(x^3-x^2\right)\)[/tex]
B. [tex]\(\log \left(x^3+x^2\right)\)[/tex]
C. [tex]\(\log \left(x^5\right)\)[/tex]
D. [tex]\(\log (x)\)[/tex]



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]