Questions: Write the expression as a single logarithm. log4(x^2-4) - 5 log4(x+2) log4(x^2-4) - 5 log4(x+2) = (Simplify your answer.)

Solution Steps

To combine the given logarithmic expression into a single logarithm, we can use the properties of logarithms. Specifically, we will use the power rule, which states that \( a \log_b(c) = \log_b(c^a) \), and the quotient rule, which states that \( \log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right) \).

1. Apply the power rule to the term \( 5 \log_4(x+2) \).
2. Use the quotient rule to combine the two logarithmic terms into a single logarithm.

We start with the expression: \[ \log_{4}(x^{2}-4) - 5 \log_{4}(x+2) \] Using the power rule of logarithms, we can rewrite \(5 \log_{4}(x+2)\) as: \[ \log_{4}((x+2)^{5}) \] Thus, the expression becomes: \[ \log_{4}(x^{2}-4) - \log_{4}((x+2)^{5}) \]

Next, we apply the quotient rule of logarithms, which states that: \[ \log_{b}(m) - \log_{b}(n) = \log_{b}\left(\frac{m}{n}\right) \] Applying this to our expression gives: \[ \log_{4}\left(\frac{x^{2}-4}{(x+2)^{5}}\right) \]

The expression can be further simplified. Noticing that \(x^{2}-4\) can be factored as \((x-2)(x+2)\), we rewrite the expression: \[ \log_{4}\left(\frac{(x-2)(x+2)}{(x+2)^{5}}\right) \] This simplifies to: \[ \log_{4}\left(\frac{x-2}{(x+2)^{4}}\right) \]

Final Answer