Natural Logarithm Calculator

Q: What is the difference between ln and log?

ln denotes the natural logarithm whose base is e , approximately 2.718. When "log" is used without further specification, it usually means the log function to base 10, although the convention can vary by field. Computer scientists often use log to mean log2, and many calculus textbooks use "log" to mean natural logarithm. If the base matters, you should check the context. In this calculator there is no ambiguity because the base of ln is always e .

Number (x)

Natural log, ln(x)

Enter a positive number to read off its natural logarithm, or type a value into the ln(x) field to compute e raised to that power. The calculator fills whichever box you leave empty.

Explore continuous growth, doubling time, and half-life

Turn a continuous rate into a doubling time, a half-life, and the time to reach any multiple.

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Show the natural log curve

Plot y = ln(x) and mark where your number lands on it.

Compare ln with log base 10 and log base 2

See the same number's logarithm in base e, 10, and 2, with the conversion factors.

The natural logarithm answers the question: To what power must e be raised to get a given number. Here, e is the irrational constant approximately equal to 2.71828. The result is written as ln (x).

This calculator allows you to calculate the natural logarithm (ln(x)) of any positive number and can also be used in reverse. If you enter a numerical value, you will get its log value. Alternatively, if you input a log value, it will give you the corresponding antilogarithm, raising e to that power to get back to the original value. You can also use this tool to directly evaluate values or solve exponential equations such as those where e raised to the power t equals 20, without having to change mode.

What is a natural logarithm?

ln (x) is log base e of x. It's another way of expressing the same fact that an exponent and a number are connected.

y = \ln(x) \quad\Longleftrightarrow\quad e^{\,y} = x

There are some important values to remember: ln(1) is 0 since e to the power of 0 is equal to 1. Ln of e is 1 because e to the power of one equals e.

Since e is between 2 and 3, you can check whether the result is plausible. As e to the fourth is about 54.6, so ln(54.6) is close to 4, which is between the value of 16 with a base of 2 and the value of 81 with a base of 3.

Why is it called natural logarithm?

This name derives from the fact that this function occurs naturally in analysis; the derivative of ln(x) is exactly 1/x, and the integral of 1/x is ln(x).

\frac{d}{dx}\ln(x) = \frac{1}{x} \qquad \int \frac{1}{x}\,dx = \ln(x) + C

If you use a different base, then there's an extra constant factor in each result. Only the base e allows both sides to remain simple. For that reason ln and its inverse function, e to the x, have become a natural language for describing any continuous change.

Where does the number value of e come from?

Jacob Bernoulli discovered the number e by studying a question about compound interest. If an amount is invested at an annual rate of interest of 100%, then in one year it will be worth 2 times as much. If the same amount is compounded half-yearly, split into two half-year periods, the result is 2.25. Compounding weekly (52 times a year) leaves us with about 2.692.

If you calculate compound interest continuously and divide it into infinitely many infinitesimally small periods, the total amount at the end of the year approaches a certain limit without increasing further. This limit is e.

e = \lim_{n\to\infty}\left(1 + \frac{1}{n}\right)^{n} \approx 2.718281828

Instructions for using this calculator:

If you enter a positive number into the field "x", then you can read off the natural logarithm of this number. Alternatively, if you enter a known logarithm in the field "ln(x)", then you can recover the original value. For example: If x = 20, then ln(20) is approximately 2.996.

If you reverse the operation, then you get the inverse function. If you put 2.996 into the ln(x) field, it will output 20 because e to the power of 2.996 is approximately 20. The second direction is the natural exponential function e to the x. So an equation of the form e to the t equals a particular number can be solved with just one input.

Converting ln to log base 10:

Because all logarithms of the same number are multiples of each other, you can convert between natural logarithms and base-10 logarithms without needing to re-compute things. The bridge is ln(10), which is approximately 2.302585.

\ln(x) = \ln(10)\,\log_{10}(x) \approx 2.302585\,\log_{10}(x)

To go in the opposite direction you just need to divide by a constant, so ln(100) is about 4.605 and if you divide 4.605 by 2.302585 you get 2 which is log base 10 of 100. The results panel shows the log base 10 and binary logarithm of the value entered.

Natural logarithms, doubling time and half-life.

The natural logarithm of two, ln(2), is not a very striking number but it is extremely useful and has the value of approximately 0.6931. It forms the basis for both doubling time and half-life calculations in radioactive substances.

If something is increasing continuously at a rate of r, then it will take ln(2) / r time to double in size and ln(M) / r time to grow by any factor M. For example, if capital grows continuously at a rate of 7%, it takes approximately ln(2)/0.07 time to double, which is about 9.9 years.

t_{\text{double}} = \frac{\ln 2}{r} \qquad t_{\times M} = \frac{\ln M}{r}

The well-known Rule of 70 is a rounded version of this formula, since ln(2) multiplied by 100 gives approximately 69.3. As the same ln(2) also determines half-life for decay, the continuous growth check box shows both doubling time and an estimate according to the rule of 70 as well as the time to reach any selected multiple.

Log identities.

Logs turn multiplication into addition, which made slide rules useful for over a century. Ln follows the same three rules.

\ln(ab) = \ln a + \ln b

\ln\!\left(\tfrac{a}{b}\right) = \ln a - \ln b

\ln(a^{\,n}) = n\,\ln a

These rules are helpful for estimating by hand calculations. Since ln(20) is the same as ln(2 × 10), you can add together ln(2) and ln(10), which are about 0.693 and 2.303 respectively. That gives 2.996, which is fairly close to the actual answer without a calculator.

Example calculation:

This rule is summarized in a short table: numbers larger than 1 have positive logarithms, the log of 1 is zero, and numbers smaller than 1 have negative logarithms.

Number x	ln(x)	Why
1	0	e to the 0 is 1
2	0.6931	the famous ln 2
e (about 2.71828)	1	e to the 1 is e
10	2.3026	ln 10, the base-10 bridge
20	2.9957	ln 2 plus ln 10
100	4.6052	twice ln 10
0.5	-0.6931	between 0 and 1, so negative

Where natural logarithms occur:

Natural logarithms appear when a quantity changes proportionally to its current size, which is the case for most situations in the continuous world.

Finance: For continuous compounding interest the formula is A = P * e^(rt), where A is the ending amount, P is the starting amount, r is the rate and t is time. The time it takes for an investment to grow to a particular multiple can be directly calculated from the natural logarithm (ln).
In biology: for a population that grows by the factor e^(kt), doubling time is the value obtained by dividing ln(2) by k, where k is a constant of growth rate.
Physics: Radioactive decay is described by the formula N = N₀ * e^(-λt), where half-life is the result of dividing the decay rate by the natural logarithm of 2.
Data science: Log transformation brings highly skewed distributions into a more symmetric form which leads to more stable graphs and regressions.
Information theory: Entropy measured in nats uses the natural logarithm; a nat is a unit that corresponds continuously to the bit, whereas for the bit the base-2 logarithm is used.
Engineering sciences: The voltage across the terminals of a capacitor during charging or discharging, as well as many cooling curves, approach a steady state in an exponential manner and can be captured using natural logarithms (ln).

Important notes:

The natural logarithm is only defined for positive numbers. ln(0) approaches negative infinity and the logarithms of negative numbers do not have real values but do exist in the complex plane, with ln(-1), for example, being equal to i * pi.

Be careful with notation. On most scientific calculators "ln" means the natural logarithm (base e), while "log" means base 10 logarithm, but in pure mathematics and software contexts "log" is often used to mean the natural logarithm. So you should check the context. Also, the product rule applies for products, not sums. ln(a + b) is not equal to ln(a) plus ln(b).

Frequently asked questions

What is the difference between ln and log?

ln denotes the natural logarithm whose base is e, approximately 2.718. When "log" is used without further specification, it usually means the log function to base 10, although the convention can vary by field. Computer scientists often use log to mean log2, and many calculus textbooks use "log" to mean natural logarithm. If the base matters, you should check the context. In this calculator there is no ambiguity because the base of ln is always e.

Why is it called a natural logarithm?

The natural logarithm appears for reasons of simplification in differential and integral calculus. The derivative of ln(x) is 1/x, and the integral of 1/x is ln(x). If one chooses a different base, there are additional constants in the respective results. Choosing the base e allows derivatives and integrals to be expressed without these extra terms. For this reason, e^x is the only function whose derivative equals itself.

Can you take the natural log of a negative number?

This is not possible in the real numbers. The natural logarithm (ln(x)) is only defined when x is greater than 0 and ln(0) approaches negative infinity. For negative numbers there is also a natural logarithm in the complex numbers. In this case, ln(-1) would be equal to i multiplied by pi, but most computers do not return that result, nor does this calculator.

How do you convert natural logarithm to base 10 logarithm?

You divide it by ln(10). The result is approximately 2.302585. So the logarithm of x to base 10 equals ln(x) divided by 2.302585. For example, ln(100) is about 4.605, and if you divide 4.605 by 2.302585, you get 2. That's the common logarithm of 100. Conversely, to go from a common logarithm to a natural logarithm, you simply multiply by that same constant.

Where does the number e come from and where is natural logarithm used?

The Euler's number e is approximately equal to 2.71828 and is the limit of (1 + 1/n)^n as n approaches infinity. It is the result of continuous compounding interest, and it is also the only base for which a function is equal to its own derivative. The natural logarithm has applications in finance, population biology, radioactive decay, data transformation, and information theory, wherever a quantity changes continuously and is proportional to its size.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Khan Academy: Intro to logarithms
Definition of logarithms, the log-exponential relationship, and the log rules.
NIST Digital Library of Mathematical Functions: Logarithms
Reference definitions and identities for the logarithm and exponential functions.