91Ó°ÊÓ

In this exercise, the first step involves applying the square root property. The square root of a number is equivalent to raising that number to the power of \[ \frac{1}{2} \]. For example, the square root of a number like \[ \sqrt{4} \] can also be expressed as \[ 4^{1/2} \]. In the context of our exercise, this concept is used to transform \[ \sqrt{9 - x^2} \] into an exponentiation form: \[ (9-x^2)^{1/2} \]. This is crucial as it sets up the problem to make use of the next property: the power rule of logarithms. Whenever you encounter square roots in logarithmic expressions, remember you can always rewrite them using the exponent of \[ \frac{1}{2} \]. This will simplify the process of solving the problem.

The power rule of logarithms is a powerful tool when simplifying logarithmic expressions. The rule states that \[ \log_a(x^b) = b \log_a(x) \]. In essence, this rule allows us to bring the exponent of a number to the front of the logarithmic expression as a coefficient. Applying this rule to our transformed expression from the square root property, we have \[ \log_a((9-x^2)^{1/2}) \]. According to the power rule, this can be rewritten as \[ \frac{1}{2} \log_a(9 - x^2) \]. This step is essential for simplifying the logarithm and making further operations more manageable. Any time you see a logarithm with an exponent inside its argument, consider using the power rule to simplify.

After applying the square root property and the power rule of logarithms, the next step often involves expressing the logarithmic expression as sums or differences. This step wasn't directly necessary in solving our specific example, \[ \log _{a} \sqrt{9-x^{2}} \], but it's a useful technique to understand. Expressions inside logarithms can often be broken down into simpler components based on the properties of logarithms. For example: \[ \log_a(xy) = \log_a(x) + \log_a(y) \] and \[ \log_a(\frac{x}{y}) = \log_a(x) - \log_a(y) \]. These properties allow us to split difficult logarithmic expressions into more manageable parts. Although not needed here, understanding how to express logarithms as sums or differences is key in more complex problems. Always look for opportunities to simplify your expressions this way.