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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]
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