What Is Natural Log (ln)?

The natural logarithm, written \(\ln(x)\), is just a logarithm with a special base: the number \(e\), which is approximately \(2.71828\). In other words:

\(\ln(x) = \log_e(x)\)

If you have not seen logarithms before, it helps to first review what a logarithm is, since ln follows exactly the same rules, just with base \(e\) instead of base 10 or base 2.

Asking "what is ln(x)?" is the same as asking "\(e\) raised to what power gives \(x\)?" For example, \(\ln(e) = 1\) because \(e^1 = e\), and \(\ln(1) = 0\) because \(e^0 = 1\).

The defining relationship between ln and the exponential function is:

\(\ln(x) = y \quad \)means\( \quad e^y = x\)

This is the same pattern used when you convert a logarithm to exponential form: the base stays the base, the log's output becomes the exponent, and the argument becomes the result. Because ln and \(e^x\) undo each other, they are inverse functions:

\(\ln(e^x) = x \quad \)and\( \quad e^{\ln(x)} = x, \; x > 0\)

You will often see two special logarithms in a textbook: the common logarithm, written \(\log(x)\), which uses base 10, and the natural logarithm, written \(\ln(x)\), which uses base \(e\). They follow the same laws of logarithms, but they show up in different places. Common logs are convenient for measuring scales like pH or decibels, while natural logs appear constantly in calculus, growth and decay models, and compound interest, because the derivative of \(e^x\) is itself, which makes \(e\) the natural choice of base.

Historically, natural logs are also called Naperian logarithms, named after John Napier, who developed early logarithm tables long before the number \(e\) was formally defined.

Because \(\ln(x)\) is only defined for \(x > 0\), its graph exists only to the right of the y-axis. As \(x\) approaches \(0\) from the right, \(\ln(x)\) drops toward negative infinity, creating a vertical asymptote at \(x = 0\). The graph crosses the x-axis at \(x = 1\), since \(\ln(1) = 0\), and it passes through \(x = e\) at a height of \(1\), since \(\ln(e) = 1\).

Notice that the curve rises slowly and keeps increasing without bound, just much more slowly than a linear or exponential function. If you need practice matching an equation to a graph like this, see finding a logarithmic function given its graph.

These identities hold for the natural log just as they do for any other base:

\(\ln(1) = 0\)

\(\ln(e) = 1\)

\(\ln(xy) = \ln(x) + \ln(y)\)

\(\ln\left(\frac{x}{y}\right) = \ln(x) - \ln(y)\)

\(\ln(x^n) = n\ln(x)\)

The product rule shown above is one of several log laws worth knowing well; you can see it worked through in detail in the lesson on the product rule of logarithms.

Evaluate \(\ln(e^3)\) without a calculator.

Since \(\ln\) and \(e^x\) are inverse operations, the exponent simply comes down: \(\ln(e^3) = 3\).

Now try \(\ln\left(\frac{1}{e^2}\right)\). Rewrite \(\frac{1}{e^2}\) as \(e^{-2}\), so \(\ln(e^{-2}) = -2\).

For practice with trickier cases that do not simplify as cleanly, work through the lesson on evaluating logs without a calculator.

Once you are comfortable with the definition and graph of ln, the next step is usually solving equations that contain it, since many real-world models (population growth, radioactive decay, continuous interest) are written using \(e\) and \(\ln\). That skill is covered separately in the lesson on how to solve logarithmic equations, where ln equations are handled using the same inverse relationship shown here: undo \(\ln\) by exponentiating both sides with base \(e\).

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