What is log(10000)- log(1/100) +log(1/10)

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: \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \).

Explore how to solve the expression log(10000) - log(1/100) + log(1/10) step-by-step with detailed explanations and examples of logarithmic properties. Suitable for high school students studying logarithms.

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