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\frac{10\log_{7}\left(7w+1\right)+\log_{7}\left(w+6\right)}{2}
Factor out \frac{1}{2}.
\frac{10\ln(7w+1)+\ln(w+6)}{\ln(7)}
Consider 10\ln(7w+1)\times \frac{1}{\ln(7)}+\ln(w+6)\times \frac{1}{\ln(7)}. Factor out \frac{1}{\ln(7)}.
\frac{10\ln(7w+1)+\ln(w+6)}{2\ln(7)}
Rewrite the complete factored expression.