Simplifying the Logarithmic Expression

The problem asks for the value of the expression:

$ E = \frac{1}{1+\log_u vw} + \frac{1}{1+\log_v wu} + \frac{1}{1+\log_w uv} $

We can simplify each term using logarithm properties. Recall that $1 = \log_b b$ for any base $b$. Let's rewrite the denominators:

• Term 1 Denominator: $1+\log_u vw = \log_u u + \log_u vw = \log_u (u \cdot vw) = \log_u (uvw)$
• Term 2 Denominator: $1+\log_v wu = \log_v v + \log_v wu = \log_v (v \cdot wu) = \log_v (uvw)$
• Term 3 Denominator: $1+\log_w uv = \log_w w + \log_w uv = \log_w (w \cdot uv) = \log_w (uvw)$

Now, substitute these back into the expression:

$ E = \frac{1}{\log_u (uvw)} + \frac{1}{\log_v (uvw)} + \frac{1}{\log_w (uvw)} $

Applying Change of Base Property

Using the logarithm property $\frac{1}{\log_a b} = \log_b a$, we can rewrite the expression:

$ E = \log_{uvw} u + \log_{uvw} v + \log_{uvw} w $

Combining Logarithms and Final Value

Using the logarithm property $\log_b x + \log_b y + \log_b z = \log_b (xyz)$, we combine the terms:

$ E = \log_{uvw} (u \cdot v \cdot w) $

$ E = \log_{uvw} (uvw) $

Finally, using the property $\log_b b = 1$:

$ E = 1 $

Therefore, the value of the expression is 1.