Lambert W Calculator: Find the Product Log

A Lambert W calculator finds the value of the Lambert W function, also called the product logarithm. Enter a real number x and choose a branch to get w satisfying w × e^w = x; for x = 1 on the principal branch, the answer is about 0.567143.

Formula

The Lambert W function is defined as the inverse of the function w ↦ w e^w. Its defining equation is:

w e^w = x, so w = W(x).

Here, x is the input number, e is the base of the natural logarithm, and w is the output. The equation alone does not always identify a unique real answer: for some x values there are two real branches. W₀(x) denotes the principal branch, while W₋₁(x) denotes the lower real branch.

The calculator specification uses the natural logarithm of x as its output expression, which is not the Lambert W function. Lambert W cannot be represented by the allowed elementary expressions in the calculator specification. Therefore, this block cannot produce a valid Lambert W result and should not be used as a working calculator. For a numerical result, use a calculator or software with a Lambert W function, or solve w e^w = x numerically.

Worked Example

For the defaults shown above, x = 1 and the principal branch is selected. The specified expression evaluates as:

w = ln(1) = 0.

But 0 × e⁰ = 0, not 1, so this is not a Lambert W answer. The actual principal-branch solution is W₀(1) ≈ 0.567143. Substitution verifies it: 0.567143 × e^0.567143 ≈ 1. This distinction matters: the displayed calculator block’s formula is unable to compute the quantity named in the title.

What Lambert W Means

Many familiar functions reverse simple operations. A square root reverses squaring, and a logarithm reverses exponentiation. Lambert W reverses a slightly more complicated operation: multiplying a number by its own exponential.

For example, if w e^w = 5, then w = W(5), approximately 1.3267. There is no ordinary algebraic rearrangement that isolates w, because it appears both outside and in the exponent. Lambert W supplies a standard name for that inverse and makes such equations easier to state and solve.

The name “product logarithm” reflects the product w e^w. It is not the same as a regular logarithm, and W(x) does not mean “W multiplied by x.” The letter W is the function name.

Branches and Real-Number Answers

For real inputs, the branch determines which solution is intended. The principal branch W₀ is defined for every real x ≥ −1/e. The lower branch W₋₁ is real only when −1/e ≤ x < 0. At x = −1/e, both branches meet at w = −1; for −1/e < x < 0, they give two distinct real answers.

• If x is positive, there is one real answer, on W₀. For instance, W₀(1) ≈ 0.567143.
• If x = 0, the real answer is W₀(0) = 0.
• If −1/e < x < 0, there are two real answers. At x = −0.1, W₀(x) is about −0.112, while W₋₁(x) is about −3.577.
• If x < −1/e, there is no real-valued answer, though complex-valued branches exist.

Thus, a calculator that accepts a real input may still need a branch setting. For a nonnegative input, the principal branch is the real choice. For a negative input in the two-solution interval, state which solution your problem requires.

How to Check a Result

A numerical Lambert W answer can be checked by putting it back into the defining equation. If the calculator reports w for input x, evaluate w e^w; it should equal x within the displayed rounding precision. For example, rounding W₀(1) to 0.567 gives 0.567e^0.567 ≈ 0.9998, close to 1.

For hand calculation or software without a Lambert W command, solve f(w) = w e^w − x = 0 with a numerical root-finding method. Newton’s method uses the update:

w_next = w − (w e^w − x) / (e^w(w + 1)).

This iteration can be sensitive near w = −1, where the denominator approaches zero. For inputs between −1/e and 0, an initial guess must also be chosen for the desired branch; otherwise an iteration may converge to the other real solution or fail. A dedicated Lambert W implementation is preferable when accuracy or branch selection matters.

Where the Function Is Used

Lambert W appears when an unknown occurs both linearly and in an exponent. It is useful for rearranging some equations in applied mathematics, physics, engineering, and computing. A common pattern is y = a + b ln(y), or an equation equivalent to y e^y = c after algebraic changes of variables.

It can also express solutions involving exponential growth or decay combined with a variable-dependent factor. In practice, recognizing the pattern can turn an equation that looks impossible to isolate into a direct expression using W. The resulting formula still requires attention to domain and branch: algebra may give multiple valid solutions, but the situation being modeled may permit only one.

Frequently Asked Questions

What does Lambert W calculate?
It returns w such that w e^w = x. It is the inverse of the product-exponential expression, not the ordinary logarithm.

What is W(1)?
The principal real value is W₀(1) ≈ 0.5671432904. It satisfies w e^w = 1.

Why can Lambert W have two answers?
For inputs between −1/e and 0, the curve y = w e^w reaches the same negative y-value once on either side of w = −1. Those roots belong to W₀ and W₋₁.

Can Lambert W be calculated with a regular logarithm?
Not in general. The equation w e^w = x has w in both the base factor and exponent, so its inverse is not expressible using ordinary logarithms alone.

When is there no real Lambert W answer?
For real-valued w, there is no solution when x < −1/e. A complex-valued solution may still exist, but it is not a real number.

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in x | Input value | - | 1 | 
sel b | Branch | Principal W₀=1, Lower W₋₁=2
out w | Lambert W value | - | ln(x) | 6