The expression \( \log\left(\frac{8x}{10}\right) \) can be simplified. Here's the step-by-step process:

### Step 1: Simplify the fraction
\[
\frac{8x}{10} = \frac{4x}{5}
\]
Now, the expression becomes:
\[
\log\left(\frac{4x}{5}\right)
\]

### Step 2: Apply the logarithm property
Using the logarithmic property \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \), we can rewrite this as:
\[
\log(4x) - \log(5)
\]

### Step 3: Apply the product rule
Next, we can apply the product rule \( \log(ab) = \log(a) + \log(b) \) to \( \log(4x) \):
\[
\log(4x) = \log(4) + \log(x)
\]
Now, substitute this back into the previous expression:
\[
\log(4) + \log(x) - \log(5)
\]

### Step 4: Combine constants
Since \( \log(4) \) and \( \log(5) \) are constants, the simplified expression becomes:
\[
\log(x) + \log(4) - \log(5)
\]

This is the fully simplified form of the expression.

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Would you like to see more details on logarithmic properties or specific examples?

Here are 5 related questions:
1. How would you simplify \( \log\left(\frac{6x}{15}\right) \)?
2. What is the value of \( \log(4) - \log(5) \) using base 10 logarithms?
3. Can you expand \( \log(100x) \) using logarithmic properties?
4. How does the change of base formula work with logarithms?
5. How do logarithmic functions behave with negative inputs?

**Tip:** Remember that logarithms are undefined for non-positive numbers like \( x \leq 0 \). Always ensure your domain excludes such values when solving equations involving logarithms.

Learn how to simplify the logarithmic expression log(8x/10) using properties such as the quotient rule and the product rule. This step-by-step guide breaks down the process, including simplification of the fraction and applying logarithmic properties. Suitable for high school students.

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