question 2 of 24 step 1 of 1 02:25:31 use the properties of logarithms to expand the following expression as…

question 2 of 24 step 1 of 1 02:25:31 use the properties of logarithms to expand the following expression as much as possible. simplify any numerical expressions that can be evaluated without a calculator. \\(\\log_{9}(81x^{2})\\) answer 2 points keypad keyboard shortcuts

question 2 of 24 step 1 of 1 02:25:31 use the properties of logarithms to expand the following expression as much as possible. simplify any numerical expressions that can be evaluated without a calculator. \\(\\log_{9}(81x^{2})\\) answer 2 points keypad keyboard shortcuts

Answer

Explanation:

Step1: Apply Product Rule of Logarithms

The product rule of logarithms states that $\log_b(MN) = \log_b(M) + \log_b(N)$. For the expression $\log_9(81x^2)$, we can split it into $\log_9(81) + \log_9(x^2)$.

Step2: Simplify $\log_9(81)$

We know that $9^2 = 81$, so by the definition of a logarithm, $\log_9(81) = 2$ because if $y = \log_b(x)$, then $b^y = x$. Here, $b = 9$, $x = 81$, and $y = 2$ since $9^2 = 81$.

Step3: Apply Power Rule of Logarithms to $\log_9(x^2)$

The power rule of logarithms states that $\log_b(M^n) = n\log_b(M)$. So, $\log_9(x^2) = 2\log_9(x)$.

Step4: Combine the Results

Putting it all together, $\log_9(81x^2) = \log_9(81) + \log_9(x^2) = 2 + 2\log_9(x)$.

Answer:

$2 + 2\log_{9}(x)$