We are given the equation:

$\log(x + 7) - \log(x) = 2\log(10)$

Step 1: Apply the properties of logarithms

We can use the logarithmic subtraction property to combine the left-hand side into a single logarithm. The property is:

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

So, the equation becomes:

$\log\left(\frac{x + 7}{x}\right) = 2\log(10)$

Step 2: Simplify the right-hand side

We can rewrite $2\log(10)$ using the logarithmic power rule:

$a\log(b) = \log(b^a)$

Thus:

$2\log(10) = \log(10^2) = \log(100)$

Now the equation is:

$\log\left(\frac{x + 7}{x}\right) = \log(100)$

Step 3: Remove the logarithms

Since the logarithms on both sides are equal, we can set the arguments of the logarithms equal to each other:

$\frac{x + 7}{x} = 100$

Step 4: Solve for $x$

Multiply both sides by $x$ to eliminate the fraction:

$x + 7 = 100x$

Subtract $x$ from both sides:

$7 = 99x$

Solve for $x$:

$x = \frac{7}{99} = \frac{1}{14}$

Step 5: Check for validity

Since the logarithm function is only defined for positive arguments, we must check that $x + 7 > 0$ and $x > 0$. With $x = \frac{1}{14}$, both conditions hold, so the solution is valid.

Final Answer:

$x = \frac{1}{14}$

Would you like further explanation or have any questions?

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Related questions:

1. How would the solution change if the right-hand side was $3\log(10)$?
2. What happens if the equation had $\log(x + 7) + \log(x)$?
3. Can this method be used to solve an equation involving natural logarithms (ln)?
4. What are the conditions for a logarithmic equation to have no solution?
5. How do the properties of logarithms help simplify complex equations?

Tip: Always check the domain of the logarithmic function when solving equations to ensure the solutions are valid.