To solve the equation 

\[
H = \frac{V - x}{\ln\left(\frac{V - r}{x - r}\right)}
\]

for \( x \) using the Lambert W function, let's proceed step by step.

### Step 1: Isolate the logarithm

First, multiply both sides by the logarithm to get rid of the fraction:

\[
H \cdot \ln\left(\frac{V - r}{x - r}\right) = V - x
\]

### Step 2: Simplify and rearrange terms

Next, we can rearrange the equation to isolate the logarithm on one side:

\[
\ln\left(\frac{V - r}{x - r}\right) = \frac{V - x}{H}
\]

### Step 3: Exponentiate both sides

Exponentiating both sides to remove the logarithm gives:

\[
\frac{V - r}{x - r} = e^{\frac{V - x}{H}}
\]

### Step 4: Isolate the term with \( x \)

Now, multiply both sides by \( (x - r) \):

\[
V - r = (x - r) \cdot e^{\frac{V - x}{H}}
\]

Next, distribute and isolate the term with \( x \) on one side:

\[
(V - r) = x \cdot e^{\frac{V - x}{H}} - r \cdot e^{\frac{V - x}{H}}
\]

\[
V - r + r \cdot e^{\frac{V - x}{H}} = x \cdot e^{\frac{V - x}{H}}
\]

### Step 5: Factor out \( x \)

To make the equation more manageable, factor out \( x \):

\[
V - r = x \cdot e^{\frac{V - x}{H}} - r \cdot e^{\frac{V - x}{H}}
\]

\[
V - r + r \cdot e^{\frac{V - x}{H}} = x \cdot e^{\frac{V - x}{H}}
\]

\[
x = \frac{V - r + r \cdot e^{\frac{V - x}{H}}}{e^{\frac{V - x}{H}}}
\]

### Step 6: Substitute \( u = \frac{V - x}{H} \)

Let \( u = \frac{V - x}{H} \), then the equation becomes:

\[
x = V - H \cdot u
\]

Substituting back into the equation:

\[
V - r + r \cdot e^u = (V - H \cdot u) \cdot e^u
\]

### Step 7: Applying the Lambert W function

We now have an equation of the form:

\[
(V - r) + r \cdot e^u = (V - H \cdot u) \cdot e^u
\]

Simplifying, we get:

\[
(V - r) = (V - H \cdot u) \cdot e^u
\]

Let \( z = \frac{(V - r)}{V} \) to simplify, giving:

\[
e^u \cdot (1 - \frac{H \cdot u}{V}) = z
\]

Rearrange the terms to use the Lambert W function:

\[
u \cdot e^u = \frac{V - r}{H \cdot u}
\]

Then the solution for \( u \) can be expressed as:

\[
u = W\left(\frac{V - r}{H \cdot u}\right)
\]

### Step 8: Solve for \( x \)

Finally, substituting back for \( x \):

\[
x = V - H \cdot W\left(\frac{V - r}{H \cdot u}\right)
\]

This gives the solution for \( x \) in terms of the Lambert W function. The exact expression for \( x \) depends on the specific values of \( V \), \( H \), and \( r \).

Let me know if you'd like any further details or have more questions!

---

**Related Questions:**
1. How can the Lambert W function be computed numerically?
2. What is the domain of the Lambert W function?
3. Can this method be applied to other logarithmic equations?
4. What is the significance of the Lambert W function in solving transcendental equations?
5. How would the equation change if \( r = 0 \)?
6. How does the solution behave as \( H \) approaches zero?
7. What are the physical interpretations of this equation in real-world contexts?
8. How do different branches of the Lambert W function affect the solution?

**Tip:** The Lambert W function is useful for solving equations where the unknown appears both in the base and the exponent, which are difficult to solve using elementary algebraic methods.

Explore how to solve the equation H = (V - x) / ln((V - r) / (x - r)) for x using the Lambert W function. This detailed solution covers the application of logarithms and Lambert's theorem, essential for advanced mathematical studies.

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