To solve the equation \( 5 \ln(4x - 10) + 11 = 1 \) for \( x \), follow these steps:

### Step 1: Isolate the logarithmic term
Start by subtracting 11 from both sides of the equation:

\[
5 \ln(4x - 10) + 11 - 11 = 1 - 11
\]

\[
5 \ln(4x - 10) = -10
\]

### Step 2: Divide by 5
Next, divide both sides of the equation by 5:

\[
\ln(4x - 10) = \frac{-10}{5}
\]

\[
\ln(4x - 10) = -2
\]

### Step 3: Exponentiate both sides
To eliminate the natural logarithm, exponentiate both sides of the equation using the base \( e \):

\[
4x - 10 = e^{-2}
\]

Since \( e^{-2} \approx 0.1353 \), we now have:

\[
4x - 10 = 0.1353
\]

### Step 4: Solve for \( x \)
Add 10 to both sides:

\[
4x = 0.1353 + 10
\]

\[
4x = 10.1353
\]

Now, divide both sides by 4:

\[
x = \frac{10.1353}{4}
\]

\[
x \approx 2.5338
\]

Thus, the solution is approximately:

\[
x \approx 2.534
\]

Would you like further details or have any questions?

Here are five related questions to expand your understanding:

1. How would you solve \( 7 \ln(3x - 8) + 12 = 5 \) for \( x \)?
2. What are the properties of logarithms that can simplify complex logarithmic expressions?
3. How does changing the base of the logarithm affect the solution to a logarithmic equation?
4. How do you solve equations involving the natural logarithm and exponential functions together?
5. What is the significance of the number \( e \) in logarithmic and exponential functions?

**Tip:** When solving logarithmic equations, always check if the argument of the logarithm stays positive after finding the solution to ensure it is valid.

Learn how to solve the equation 5ln(4x-10)+11=1 for x with a detailed step-by-step solution. This problem involves logarithms and exponential equations, suitable for high school students in Grades 10-12.

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