log (x    –    13  )    +    3    log      2    =    log      (   3x   + 1)

To solve the equation \( \log(x - 13) + 3\log(2) = \log(3x + 1) \), we can follow these steps:

### Step 1: Simplify the equation
We know from logarithmic properties that \( a \log(b) = \log(b^a) \). So, we can rewrite \( 3\log(2) \) as \( \log(2^3) \).

\[
\log(x - 13) + \log(2^3) = \log(3x + 1)
\]

Since \( 2^3 = 8 \), the equation becomes:

\[
\log(x - 13) + \log(8) = \log(3x + 1)
\]

### Step 2: Combine the logarithms on the left-hand side
Using the logarithmic property \( \log(a) + \log(b) = \log(a \times b) \), we can combine the logarithms:

\[
\log((x - 13) \times 8) = \log(3x + 1)
\]

Simplify the left-hand side:

\[
\log(8x - 104) = \log(3x + 1)
\]

### Step 3: Remove the logarithms
If \( \log(a) = \log(b) \), then \( a = b \). Therefore, we can equate the arguments of the logarithms:

\[
8x - 104 = 3x + 1
\]

### Step 4: Solve the linear equation
Subtract \( 3x \) from both sides:

\[
5x - 104 = 1
\]

Add 104 to both sides:

\[
5x = 105
\]

Divide by 5:

\[
x = 21
\]

### Step 5: Check the solution
Substitute \( x = 21 \) back into the original logarithmic expressions to ensure they are defined and valid:

- \( x - 13 = 21 - 13 = 8 \) (which is positive, so valid for logarithms).
- \( 3x + 1 = 3(21) + 1 = 63 + 1 = 64 \) (also positive).

Since both arguments are positive, \( x = 21 \) is a valid solution.

### Final Answer:
The solution to the equation is \( x = 21 \).

Would you like further details on any of these steps or have additional questions?

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Here are 5 related questions that could expand your understanding:

1. How do logarithmic properties help simplify equations?
2. What happens if one side of the logarithmic equation is negative?
3. How do you solve an equation involving logarithms with different bases?
4. What is the significance of ensuring that logarithmic arguments are positive?
5. Can you have a logarithmic equation where no solution exists?

**Tip:** Always check that the arguments of logarithmic functions remain positive when solving equations.

Learn how to solve the logarithmic equation log(x - 13) + 3log(2) = log(3x + 1) with a detailed step-by-step solution. This problem involves logarithmic properties and equations, suitable for high school students (Grades 10-12).

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