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The equation to solve is:

\[
\ln(x + 5) = 1 + \ln(x)
\]

We want to express the solution in terms of \( e \). Here’s how we can solve it step by step:

### Step 1: Use logarithmic properties
We can start by isolating the logarithmic terms. Subtract \( \ln(x) \) from both sides:

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

Using the logarithmic property \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \), we simplify the left side:

\[
\ln\left(\frac{x + 5}{x}\right) = 1
\]

### Step 2: Eliminate the natural logarithm
To eliminate the natural logarithm, we exponentiate both sides, using the fact that \( e^{\ln(a)} = a \):

\[
\frac{x + 5}{x} = e^1 = e
\]

### Step 3: Solve the equation
Now we solve the equation:

\[
\frac{x + 5}{x} = e
\]

Multiply both sides by \( x \):

\[
x + 5 = ex
\]

Now subtract \( x \) from both sides:

\[
5 = ex - x
\]

Factor out \( x \) on the right-hand side:

\[
5 = x(e - 1)
\]

Finally, solve for \( x \):

\[
x = \frac{5}{e - 1}
\]

### Final Answer
The solution is:

\[
x = \frac{5}{e - 1}
\]

Would you like further details or have any questions?

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Here are 5 related questions:

1. What properties of logarithms did we use in this solution?
2. How do you exponentiate both sides of an equation to eliminate a logarithm?
3. Why is the base of the natural logarithm \( e \)?
4. What would happen if the logarithms were base 10 instead of natural logarithms?
5. How would you solve \( \ln(x + 2) = 2 + \ln(x) \)?

**Tip:** When solving logarithmic equations, always check if the arguments of logarithms stay positive for the values of \( x \).

Solve the equation ln(x + 5) = 1 + ln(x), giving your answer in terms of e.

Learn how to solve the equation ln(x + 5) = 1 + ln(x) by using logarithmic properties and exponential functions. This solution breaks down each step and provides the final answer in terms of e. Suitable for high school students in Grades 11-12.

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