Sure! Let's solve [tex]\(\log_3\left(\frac{9}{6}\right)\)[/tex] step-by-step.
1. Simplify the fraction inside the logarithm:
[tex]\[
\frac{9}{6} = \frac{3 \cdot 3}{3 \cdot 2} = \frac{3}{2}
\][/tex]
So, we need to find:
[tex]\[
\log_3\left(\frac{3}{2}\right)
\][/tex]
2. Understand what the logarithm represents:
A logarithm in the form [tex]\(\log_b(a)\)[/tex] is the power to which the base [tex]\(b\)[/tex] must be raised to obtain the number [tex]\(a\)[/tex]. In this case, we are looking for the exponent [tex]\(x\)[/tex] such that:
[tex]\[
3^x = \frac{3}{2}
\][/tex]
3. Using properties of logarithms:
To solve for [tex]\(x\)[/tex], you would normally use the change of base formula or any logarithm properties, but here we present the final evaluated solution directly:
So, the value of [tex]\(\log_3\left(\frac{9}{6}\right)\)[/tex] or [tex]\(\log_3\left(\frac{3}{2}\right)\)[/tex] approximately equals to:
[tex]\[
0.3690702464285425
\][/tex]
Thus,
[tex]\[
\boxed{0.3690702464285425}
\][/tex]
