Logarithm Expression Simplification

To determine the value of the given mathematical expression, we will systematically simplify each of its terms by applying fundamental logarithm properties. The expression we need to evaluate is:

\[ \frac{1}{{1 + {{\log }_u}vw}} + \frac{1}{{1 + {{\log }_v}wu}} + \frac{1}{{1 + {{\log }_w}uv}} \]

Logarithm Properties Used

The solution relies on the following essential logarithm properties:

• Logarithm Identity: For any base \(b > 0, b \ne 1\), we have \(\log_b b = 1\). This property allows us to express the number 1 in terms of a logarithm with a specific base.
• Logarithm Product Rule: For positive numbers \(M\) and \(N\) and a base \(b > 0, b \ne 1\), \(\log_b M + \log_b N = \log_b (MN)\). This rule helps combine sums of logarithms into a single logarithm of a product.
• Change of Base Formula (Reciprocal Form): For positive numbers \(a\) and \(b\) (where \(a, b \ne 1\)), \(\frac{1}{{\log_b a}} = \log_a b\). This property is crucial for changing the base of a logarithm and simplifying fractions involving logarithms.

Simplifying Each Term of the Logarithm Expression

Term 1: Simplifying \(\frac{1}{{1 + {{\log }_u}vw}}\)

Let's focus on the denominator of the first term:

\[ 1 + {{\log }_u}vw \]

Using the logarithm identity, we can rewrite \(1\) as \(\log_u u\). Substituting this into the denominator:

\[ \log_u u + {{\log }_u}vw \]

Now, apply the logarithm product rule \(\log_b M + \log_b N = \log_b (MN)\). Here, \(M=u\) and \(N=vw\):

\[ \log_u (u \cdot vw) = \log_u (uvw) \]

So, the first term of the expression becomes:

\[ \frac{1}{{\log_u (uvw)}} \]

Finally, applying the change of base formula \(\frac{1}{{\log_b a}} = \log_a b\), where \(b=u\) and \(a=uvw\), we get:

\[ \log_{uvw} u \]

Term 2: Simplifying \(\frac{1}{{1 + {{\log }_v}wu}}\)

We follow a similar process for the second term's denominator:

\[ 1 + {{\log }_v}wu \]

Replace \(1\) with \(\log_v v\) using the logarithm identity:

\[ \log_v v + {{\log }_v}wu \]

Apply the logarithm product rule \(\log_b M + \log_b N = \log_b (MN)\):

\[ \log_v (v \cdot wu) = \log_v (uvw) \]

Thus, the second term of the expression is:

\[ \frac{1}{{\log_v (uvw)}} \]

Using the change of base formula \(\frac{1}{{\log_b a}} = \log_a b\), where \(b=v\) and \(a=uvw\), we obtain:

\[ \log_{uvw} v \]

Term 3: Simplifying \(\frac{1}{{1 + {{\log }_w}uv}}\)

And for the third term's denominator:

\[ 1 + {{\log }_w}uv \]

Substitute \(1\) with \(\log_w w\) using the logarithm identity:

\[ \log_w w + {{\log }_w}uv \]

Apply the logarithm product rule \(\log_b M + \log_b N = \log_b (MN)\):

\[ \log_w (w \cdot uv) = \log_w (uvw) \]

Therefore, the third term of the expression is:

\[ \frac{1}{{\log_w (uvw)}} \]

Applying the change of base formula \(\frac{1}{{\log_b a}} = \log_a b\), where \(b=w\) and \(a=uvw\), we get:

\[ \log_{uvw} w \]

Combining Simplified Logarithm Terms

Now, we substitute these simplified forms back into the original expression:

\[ \log_{uvw} u + \log_{uvw} v + \log_{uvw} w \]

Since all terms now have the same base, \(uvw\), we can apply the logarithm product rule multiple times:

\[ \log_{uvw} (u \cdot v \cdot w) \]

This simplifies to:

\[ \log_{uvw} (uvw) \]

Final Logarithm Expression Value

Finally, using the logarithm identity \(\log_b b = 1\), where the base and the argument of the logarithm are identical (in this case, both are \(uvw\)), we conclude:

\[ \log_{uvw} (uvw) = 1 \]

Thus, the value of the given logarithm expression is 1.