Solve the equation. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)
log(x + 7) − log x = 2 log 10

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?

---

**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.

Solve the equation. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) log(x + 7) − log x = 2 log 10

Learn how to solve the logarithmic equation log(x + 7) − log x = 2 log 10 step-by-step using properties of logarithms. This problem involves applying the logarithmic subtraction rule and solving for x, making it ideal for high school students in Grades 9-12.

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