Question

Write the expression as the logarithm of a single quantity. $$ 2 \ln 3-\frac{1}{2} \ln \left(x^{2}+1\right) $$

Short answer

The simplified form of the given expression as the logarithm of a single quantity is \( \ln (9\sqrt{x^2+1}) \).

Step-by-step solution

Apply the power rule to the logarithms

The first part of the given expression \(2 \ln 3\) can be written as \( \ln (3^2) \) using the power rule. Similarly for the second part, \(-\frac{1}{2} \ln (x^2+1)\), the power rule can be applied to produce \( \ln ((x^2+1)^{-1/2}) \). That makes the expression look like \( \ln (3^2) - \ln ((x^2+1)^{-1/2}) \).

Apply the quotient rule to the logarithms

After applying the power rule, the expression now consists of a subtraction operation. This subtraction operation can be rewritten as division inside of a single logarithm using the quotient rule. Applying this rule yields \( \ln (3^2 / (x^2+1)^{-1/2}) \).

Simplify the expression

Finally, simplify the expression in the logarithm. The expression \(3^2 / (x^2+1)^{-1/2}\) simplifies to \(9 \cdot ((x^2+1)^{1/2})\). The expression now becomes \( \ln (9\sqrt{x^2+1}) \).