How does this bonding curve work?

The formula y = a ln(x) + b defines a logarithmic bonding curve where the token price increases quickly at first and then slows down as more tokens are bought. Here, x represents the current token supply, a controls how steep the curve is, and b is the base price.

When x is small, the logarithm grows rapidly, so the price increases quickly — early buyers experience noticeable price movement even with small purchases. As x becomes larger, the growth of ln(x) slows down, causing the price to increase more gradually over time.

This creates an early acceleration effect followed by stabilization, meaning early buyers still benefit, but the system becomes less aggressive and more stable as supply grows.

Advantages and limitations

One of the main advantages of this model is its smooth and natural growth pattern. It avoids extreme spikes and becomes more stable over time, making it suitable for long-term sustainability and better user experience.

However, the logarithmic model increases price very quickly at the beginning, which can discourage very early participation if a is too high. Additionally, like other bonding curves, it does not reflect real market demand, meaning the price is entirely determined by the formula rather than external factors.