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.