- Rewrite the square root as an exponent: \(\frac{1}{2} \log \left( \frac{x^2+6}{(x^2+7)(x^3-2)^2} \right)\).
- Apply the quotient rule: \(\frac{1}{2} [ \log(x^2+6) - \log((x^2+7)(x^3-2)^2) ]\).
- Apply the product rule and distributive property: \(\frac{1}{2} [ \log(x^2+6) - \log(x^2+7) - \log((x^3-2)^2) ]\).
- Simplify using the power rule to arrive at the final result: \(\boxed{\frac{1}{2} \log(x^2+6) - \frac{1}{2} \log(x^2+7) - \log(x^3-2)}\).
Explanation
- Analyzing the Logarithmic Expression
To expand the logarithmic expression \(\log \left( \sqrt{\frac{x^2+6}{\left(x^2+7\right)\left(x^3-2\right)^2}} \right)\), we first recall the essential properties of logarithms:
- Power Rule: \(\log(A^n) = n\log(A)\)
- Quotient Rule: \(\log\left(\frac{A}{B}\right) = \log(A) - \log(B)\)
- Product Rule: \(\log(AB) = \log(A) + \log(B)\)
- Radical Rule: \(\sqrt[n]{A} = A^{1/n}\)
Using these properties, we can systematically break down the given expression.
- Applying the Power Rule for Radicals
First, we express the square root as an exponent:
\[ \log \left( \left[ \frac{x^2+6}{(x^2+7)(x^3-2)^2} \right]^{1/2} \right) \]
Applying the power rule, where the exponent \(n = \frac{1}{2}\) moves to the front:
\[ \frac{1}{2} \log \left( \frac{x^2+6}{(x^2+7)(x^3-2)^2} \right) \]
- Applying Quotient and Product Rules
Now, apply the quotient rule \(\log(\frac{A}{B}) = \log(A) - \log(B)\) to the expression inside the logarithm:
\[ \frac{1}{2} \left[ \log(x^2+6) - \log((x^2+7)(x^3-2)^2) \right] \]
Next, apply the product rule \(\log(AB) = \log(A) + \log(B)\) to the second term inside the brackets:
\[ \frac{1}{2} \left[ \log(x^2+6) - (\log(x^2+7) + \log((x^3-2)^2)) \right] \]
Distribute the negative sign:
\[ \frac{1}{2} \left[ \log(x^2+6) - \log(x^2+7) - \log((x^3-2)^2) \right] \]
- Final Expansion and Simplification
Finally, apply the power rule \(\log(A^n) = n\log(A)\) to the term \(\log((x^3-2)^2)\), which becomes \(2\log(x^3-2)\). Substituting this back:
\[ \frac{1}{2} \left[ \log(x^2+6) - \log(x^2+7) - 2\log(x^3-2) \right] \]
Distributing the \(\frac{1}{2}\) throughout the expression:
\[ \frac{1}{2} \log(x^2+6) - \frac{1}{2} \log(x^2+7) - \log(x^3-2) \]
This is the fully expanded form.
Examples
Logarithms are widely used in fields like acoustics and chemistry to manage large variations in values, such as measuring sound intensity in decibels or determining the pH of chemical solutions. Expanding complex logarithmic expressions helps in simplifying these calculations by transforming complicated products and powers into manageable sums and differences of individual logarithmic terms. For instance, in data analysis, logarithmic scales are frequently used to visualize growth rates and trends that span several orders of magnitude.