When Logarithms Start Lying: The Complex Log and Its Many Truths

🌀 When Logarithms Start Lying: The Complex Log and Its Many Truths

Everyone trusts the logarithm — until it goes complex.

We’re taught that

log(xy)=log(x)+log(y)

and it works beautifully… until one day, you try it with negative numbers and suddenly

log(-1)=iπ=i(π+2π),

which is both true and false — depending on which truth you pick.

Welcome to the complex logarithm: the polite liar of mathematics.

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🎭 One Function, Infinite Personalities

In the real world, the logarithm is simple: it tells us how many times to multiply a base to reach a number.
In the complex plane, it keeps a secret: numbers don’t just stretch — they rotate.

If

z=re^iθ

then

log(z)=ln(r)+iθ

But here’s the twist: you can always add any integer multiple of 2π to θ and still describe the same point.

So instead of one value, the complex logarithm has infinitely many — each differing by 2πi.

That’s not an error. It’s geometry whispering that rotation and repetition are built into the fabric of the complex plane.

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✂️ The Need for Branch Cuts

To make sense of this infinity, mathematicians do something wonderfully inelegant:
they cut the plane.

We choose a line (usually the negative real axis) and declare:

“Here, and only here, we refuse to be continuous.”

That’s called a branch cut.
It’s an arbitrary scar we draw to keep the function single-valued elsewhere.

So the complex logarithm isn’t truly a function — it’s a patchwork of infinitely many consistent but overlapping stories.

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🔁 The Infinite Staircase

Visualize the logarithm’s surface as an endless spiral staircase winding upward around the origin.
Each time you make a full rotation around zero, you move up by 2πi.

It’s not just a metaphor — this is the Riemann surface of the logarithm.
A beautiful geometric object that makes multi-valuedness look inevitable rather than problematic.

On that surface, the logarithm becomes single-valued again.
Infinity finds its order by rising in layers.

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⚙️ Where Analysis Learns to Breathe

The complex logarithm is more than a curiosity — it’s a gateway to analytic continuation, the idea that a function can be extended beyond its apparent domain by consistency, not by formula.

That same principle underlies the extension of the Riemann zeta function, whose behavior on the critical line defines the Riemann Hypothesis.

In both cases, mathematics says:

“Don’t redefine. Continue the story.”

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📘 A Note from The Riemann Hypothesis Revealed

In my book, The Riemann Hypothesis Revealed: A Comprehensive Guide Through Complex Analysis, I explore how functions like the logarithm and zeta transcend their initial definitions — becoming richer and more unified when seen through the lens of analytic continuation. https://amzn.eu/d/1Cn2cT4

The complex logarithm transforms shapes in a beautifully systematic way.
A sector in the z-plane — that is, all points with radii between r₁ and r₂ and angles between θ₁ and θ₂ — is mapped by the logarithm into a rectangular strip in the w-plane.

Under the map w = log(z) = ln(r) + iθ:

• The circular arcs (r = constant) become vertical lines at ln(r₁) and ln(r₂).
• The radial lines (θ = constant) become horizontal lines at iθ₁ and iθ₂.

So multiplication (scaling) in the z-plane becomes translation in the real direction,
and rotation becomes translation in the imaginary direction.

The complex logarithm literally straightens out the polar grid — turning spirals into lines and sectors into strips.

 

The complex logarithm, with all its apparent contradictions, is not an embarrassment to be fixed — it’s a model for how mathematics grows by embracing ambiguity and then organizing it.

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🤔 Final Thought

If real logarithms are accountants — precise, tidy, and predictable — then complex logarithms are jazz musicians:
improvising within structure, repeating themes with variation, and occasionally adding  just for fun.