Let's expand the given logarithmic expression [tex]\(\log_b(x^3)\)[/tex] step-by-step using properties of logarithms:
1. Identify the rule applicable: The given expression is [tex]\(\log_b(x^3)\)[/tex]. For logarithmic expressions involving exponents, we can use the power rule of logarithms. The power rule states that [tex]\(\log_b(x^n) = n \cdot \log_b(x)\)[/tex].
2. Apply the power rule: In our expression, the exponent [tex]\(n\)[/tex] is 3. By applying the power rule, we can rewrite the expression as:
[tex]\[
\log_b(x^3) = 3 \cdot \log_b(x)
\][/tex]
Therefore, the expanded logarithmic expression is:
[tex]\[
3 \cdot \log_b(x)
\][/tex]
In conclusion, [tex]\(\log_b(x^3)\)[/tex] simplifies to [tex]\(3 \cdot \log_b(x)\)[/tex] when expanded using the properties of logarithms.
