solve for this with step by step solution with laws
a. log 15x = log 30
b. log (3x-2)=2
c. log 9 + log (x-8)=4
d. log (x+1) = log 2x
e. ln (x+2) -ln (x-7) = ln 6

Let's solve each equation step by step, using logarithmic laws. I'll explain the key steps along the way.

### Problem (a): 
\[
\log(15x) = \log(30)
\]

**Solution:**
1. Since \(\log(15x) = \log(30)\), we can equate the arguments because the logs are equal:
   \[
   15x = 30
   \]
2. Divide both sides by 15:
   \[
   x = \frac{30}{15} = 2
   \]
3. **Solution:** \( x = 2 \).

### Problem (b): 
\[
\log(3x - 2) = 2
\]

**Solution:**
1. Rewrite the equation in exponential form (using the fact that if \(\log_b(y) = a\), then \(y = b^a\)):
   \[
   3x - 2 = 10^2
   \]
2. Simplify \(10^2\) to get:
   \[
   3x - 2 = 100
   \]
3. Add 2 to both sides:
   \[
   3x = 102
   \]
4. Divide by 3:
   \[
   x = \frac{102}{3} = 34
   \]
5. **Solution:** \( x = 34 \).

### Problem (c): 
\[
\log 9 + \log(x - 8) = 4
\]

**Solution:**
1. Use the **product rule** of logarithms: \(\log a + \log b = \log(ab)\).
   \[
   \log(9(x - 8)) = 4
   \]
2. Rewrite in exponential form:
   \[
   9(x - 8) = 10^4
   \]
3. Calculate \(10^4 = 10000\):
   \[
   9(x - 8) = 10000
   \]
4. Divide by 9:
   \[
   x - 8 = \frac{10000}{9}
   \]
5. Add 8 to both sides:
   \[
   x = \frac{10000}{9} + 8
   \]
6. **Solution:** \( x = \frac{10000}{9} + 8 \).

### Problem (d): 
\[
\log(x + 1) = \log(2x)
\]

**Solution:**
1. Since \(\log(x + 1) = \log(2x)\), equate the arguments:
   \[
   x + 1 = 2x
   \]
2. Rearrange the equation:
   \[
   1 = x
   \]
3. **Solution:** \( x = 1 \).

### Problem (e): 
\[
\ln(x + 2) - \ln(x - 7) = \ln 6
\]

**Solution:**
1. Use the **quotient rule** of logarithms: \(\ln a - \ln b = \ln\left(\frac{a}{b}\right)\).
   \[
   \ln\left(\frac{x + 2}{x - 7}\right) = \ln 6
   \]
2. Since \(\ln\left(\frac{x + 2}{x - 7}\right) = \ln 6\), equate the arguments:
   \[
   \frac{x + 2}{x - 7} = 6
   \]
3. Cross-multiply:
   \[
   x + 2 = 6(x - 7)
   \]
4. Expand the right side:
   \[
   x + 2 = 6x - 42
   \]
5. Rearrange to solve for \(x\):
   \[
   2 + 42 = 6x - x
   \]
   \[
   44 = 5x
   \]
6. Divide by 5:
   \[
   x = \frac{44}{5} = 8.8
   \]
7. **Solution:** \( x = 8.8 \).

---

Let me know if you need further details on any step!

#### Related Questions:
1. What are the common logarithmic properties used in problem-solving?
2. How can you verify solutions in logarithmic equations?
3. What happens if there is no solution for a logarithmic equation?
4. What is the difference between logarithmic and exponential equations?
5. Can logarithmic equations have complex solutions?

#### Tip:
Always check the domain of the variables in logarithmic equations to avoid undefined expressions (e.g., arguments of logs must be positive).

Solve for this with step-by-step solution with laws:

a. log 15x = log 30
b. log (3x-2) = 2
c. log 9 + log (x-8) = 4
d. log (x+1) = log 2x
e. ln (x+2) - ln (x-7) = ln 6

Learn to solve logarithmic equations with step-by-step solutions and logarithmic laws. This guide covers various types of problems, including equations with equal logarithmic arguments, product and quotient rules, and converting logarithmic forms to exponential forms. Perfect for Grades 10-12.

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