\frac{1}{1+\log _a\left(bc\right)}+\frac{1}{1+\log _b\left(ac\right)}+\frac{1}{1+\log _c\left(ab\right)}

asked by guest
on Nov 14, 2024 at 6:54 pm



You asked:

Evaluate the expression: \(\frac{1}{1 + \log_{a}\left( b c \right)} + \frac{1}{1 + \log_{b}\left( a c \right)} + \frac{1}{1 + \log_{c}\left( a b \right)}\)

MathBot Answer:

Evaluated



\(\displaystyle \frac{1}{1 + \log_{a}\left( b c \right)} + \frac{1}{1 + \log_{b}\left( a c \right)} + \frac{1}{1 + \log_{c}\left( a b \right)} = \frac{1}{1 + \frac{\log{\left(a b \right)}}{\log{\left(c \right)}}} + \frac{1}{1 + \frac{\log{\left(a c \right)}}{\log{\left(b \right)}}} + \frac{1}{1 + \frac{\log{\left(b c \right)}}{\log{\left(a \right)}}} \)

Expanded

\[\frac{1}{1 + \log_{a}\left( b c \right)} + \frac{1}{1 + \log_{b}\left( a c \right)} + \frac{1}{1 + \log_{c}\left( a b \right)} = \frac{1}{1 + \frac{\log{\left(a b \right)}}{\log{\left(c \right)}}} + \frac{1}{1 + \frac{\log{\left(a c \right)}}{\log{\left(b \right)}}} + \frac{1}{1 + \frac{\log{\left(b c \right)}}{\log{\left(a \right)}}}\]

Factored

\[\frac{1}{1 + \log_{a}\left( b c \right)} + \frac{1}{1 + \log_{b}\left( a c \right)} + \frac{1}{1 + \log_{c}\left( a b \right)} = \frac{3 + \frac{2 \log{\left(a b \right)}}{\log{\left(c \right)}} + \frac{2 \log{\left(a c \right)}}{\log{\left(b \right)}} + \frac{\log{\left(a b \right)} \log{\left(a c \right)}}{\log{\left(b \right)} \log{\left(c \right)}} + \frac{2 \log{\left(b c \right)}}{\log{\left(a \right)}} + \frac{\log{\left(a b \right)} \log{\left(b c \right)}}{\log{\left(a \right)} \log{\left(c \right)}} + \frac{\log{\left(a c \right)} \log{\left(b c \right)}}{\log{\left(a \right)} \log{\left(b \right)}}}{\left(1 + \frac{\log{\left(b c \right)}}{\log{\left(a \right)}}\right) \left(1 + \frac{\log{\left(a c \right)}}{\log{\left(b \right)}}\right) \left(1 + \frac{\log{\left(a b \right)}}{\log{\left(c \right)}}\right)}\]

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