The expression provided is:

$\log_a x^3 - \log_a x$

Simplification Steps:

We will use the logarithmic properties:

1. Power Rule: $\log_a x^n = n \cdot \log_a x$
2. Difference Rule: $\log_a x - \log_a y = \log_a \frac{x}{y}$

Step 1: Apply the power rule to $\log_a x^3$

$\log_a x^3 = 3 \cdot \log_a x$

Substitute this back into the expression: $3 \cdot \log_a x - \log_a x$

Step 2: Simplify the coefficients

$(3 - 1) \cdot \log_a x = 2 \cdot \log_a x$

Final Answer:

$\log_a x^3 - \log_a x = 2 \cdot \log_a x$

Would you like me to explain any step in more detail or solve a related problem?

Related Questions:

1. How does the power rule for logarithms work in other cases, like $\log_a (x^n)$?
2. Can you expand the subtraction rule for three terms, like $\log_a x^4 - \log_a x^2 - \log_a x$?
3. What happens if $a = 10$ (common logarithm) or $a = e$ (natural logarithm) in such expressions?
4. Can this simplification be represented graphically? If yes, what does it look like?
5. What are other real-world applications of logarithmic simplifications like this?

Tip:

Always simplify logarithmic expressions by applying the power, product, or quotient rules first to make calculations easier!