The value of the expression 

Correct Answer: 1

The image shows the following mathematical expression containing logarithmic terms:

$\frac{1}{1+{\log }_u\left(vw\right)}+\frac{1}{1+{\log }_v\left(wu\right)}+\frac{1}{1+{\log }_w\left(uv\right)}$

We can simplify each term in the sum one by one using standard properties of logarithms.

Step 1: Simplify the first term

Recall that the number 1 can be expressed as a logarithm where the base and the argument are the same:

$1={\log }_{u}u$

Substituting this representation of 1 into the denominator of the first term gives:

$1+{\log }_u\left(vw\right)={\log }_{u}u+{\log }_u\left(vw\right)$

Applying the logarithmic product rule, ${\log }_{b}x+{\log }_{b}y={\log }_b\left(xy\right)$, we can combine the terms:

${\log }_{u}u+{\log }_u\left(vw\right)={\log }_u\left(uvw\right)$

Now, we use the change-of-base reciprocal identity, $\frac{1}{{\log }_{a}b}={\log }_{b}a$, to rewrite the entire first term:

$\frac{1}{1+{\log }_u\left(vw\right)}=\frac{1}{{\log }_u\left(uvw\right)}={\log }_{uvw}u$

Step 2: Simplify the second and third terms

Following the same method for the second term:

$1+{\log }_v\left(wu\right)={\log }_{v}v+{\log }_v\left(wu\right)={\log }_v\left(uvw\right)$

Taking its reciprocal:

$\frac{1}{1+{\log }_v\left(wu\right)}=\frac{1}{{\log }_v\left(uvw\right)}={\log }_{uvw}v$

Similarly, for the third term:

$1+{\log }_w\left(uv\right)={\log }_{w}w+{\log }_w\left(uv\right)={\log }_w\left(uvw\right)$

Taking its reciprocal:

$\frac{1}{1+{\log }_w\left(uv\right)}=\frac{1}{{\log }_w\left(uvw\right)}={\log }_{uvw}w$

Step 3: Combine all terms

Substitute the simplified parts back into the original expression:

${\log }_{uvw}u+{\log }_{uvw}v+{\log }_{uvw}w$

Use the product rule to combine the sum of logarithms with a common base:

${\log }_{uvw}u+{\log }_{uvw}v+{\log }_{uvw}w={\log }_{uvw}(u\cdot v\cdot w)={\log }_{uvw}\left(uvw\right)$

Since the argument is equal to the base of the logarithm, the value is simply 1:

${\log }_{uvw}\left(uvw\right)=1$

Thus, the value of the expression is 1.