To solve the equation

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

for \(x\) using the Lambert W function, we'll proceed with the following steps:

### Step 1: Rearrange the equation
Start by multiplying both sides by the logarithm term:

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

Next, isolate the logarithmic term:

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

### Step 2: Exponentiate both sides
Exponentiate both sides to remove the logarithm:

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

### Step 3: Cross-multiply and simplify
Cross-multiply to get rid of the fraction:

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

Rearrange the equation to group the terms involving \(x\) together:

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

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

Factor out \(x\) from the right side:

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

### Step 4: Isolate the exponential term
Isolate the exponential term involving \(x\):

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

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

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

### Step 5: Introduce the Lambert W function
To introduce the Lambert W function, we aim to express the equation in the form \(y = ze^z\), which is the definition of the Lambert W function. Set \(z = \frac{x - V}{H}\) and rearrange:

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

\[
W\left(\frac{r - V}{H} e^{\frac{r - V}{H}}\right)
\]

Thus, solving for \(x\), we get:

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

This is the solution for \(x\) in terms of the Lambert W function.

---

Do you want details or have any questions? Here are 8 related questions:

1. Would you like a detailed explanation of the Lambert W function and its properties?
2. Do you need a specific example with numerical values to see how to apply this solution?
3. Are you interested in the different branches of the Lambert W function and how they affect the solution?
4. Would you like to know the conditions under which this solution is valid?
5. Do you need help with any specific steps in the derivation?
6. Are you curious about the graphical interpretation of the solution?
7. Would you like to explore other methods to solve the equation without using the Lambert W function?
8. Do you need this solution to be applied in a specific context or problem?

**Tip:** The Lambert W function has multiple branches, but the principal branch (W₀) is the one most commonly used for real-valued solutions.

Learn how to solve equations involving the Lambert W function step-by-step. This problem covers advanced mathematical concepts such as equations, logarithms, exponentials, and the Lambert W function. Suitable for advanced mathematics enthusiasts and professionals.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation