Let's solve the expression $\log(10000) - \log\left(\frac{1}{100}\right) + \log\left(\frac{1}{10}\right)$.

Step 1: Simplify each logarithmic term.

1. Evaluate $\log(10000)$: $\log(10000) = \log(10^4) = 4 \cdot \log(10) = 4$ (Since $\log(10) = 1$)

2. Evaluate $\log\left(\frac{1}{100}\right)$: $\log\left(\frac{1}{100}\right) = \log\left(10^{-2}\right) = -2 \cdot \log(10) = -2$

3. Evaluate $\log\left(\frac{1}{10}\right)$: $\log\left(\frac{1}{10}\right) = \log\left(10^{-1}\right) = -1 \cdot \log(10) = -1$

Step 2: Substitute the values into the original expression.

$\log(10000) - \log\left(\frac{1}{100}\right) + \log\left(\frac{1}{10}\right) = 4 - (-2) + (-1)$

Step 3: Simplify the expression.

$4 + 2 - 1 = 5$

So, the value of the expression is 5.

Would you like more details or have any questions? Here are some related problems you might find interesting:

1. What is $\log(1000) - \log(10) + \log(100)$?
2. How do you simplify $\log\left(\frac{a}{b}\right)$ using logarithmic properties?
3. What is the value of $\log(1) + \log(10)$?
4. If $\log(x) = 3$, what is $x$?
5. How does the change of base formula for logarithms work?

Tip: Remember, the logarithm of a quotient can be expressed as the difference of the logarithms: $\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$.