Question - Combining Logarithmic Expressions into a Single Logarithm

The expression given is \( \ln(7) + 4\ln(4) \).

First, use the logarithm power rule \( a\ln(b) = \ln(b^a) \) to rewrite \( 4\ln(4) \) as \( \ln(4^4) \).

Thus, \( 4\ln(4) = \ln(4^4) = \ln(256) \).

Now, use the logarithm addition rule \( \ln(a) + \ln(b) = \ln(a \cdot b) \) to combine \( \ln(7) \) and \( \ln(256) \) into a single logarithm.

\( \ln(7) + \ln(256) = \ln(7 \cdot 256) = \ln(1792) \).

Therefore, \( \ln(7) + 4\ln(4) \) can be expressed as a single logarithm \( \ln(1792) \).