Free Log Calculator

The Log Calculator is a free log calculator online that computes the logarithm of any positive real number for a base you choose (the base must be positive and not equal to 1). Whether you need a natural log calculator, a log base 2 calculator, or a standard base‑10 logarithm, this logarithm calculator delivers an immediate answer. It also uses the change‑of‑base formula internally, so it works seamlessly for any base you specify.

What is a logarithm?

A logarithm is the inverse operation of exponentiation. If $b^{y} = x$ with $b > 0$ and $b \neq 1$, then the logarithm of $x$ with base $b$ is $y$: $\log_{b}(x) = y$. In other words, $\log_{b}(x)$ tells you the exponent to which $b$ must be raised to obtain $x$. For example, $2^{3} = 8$ implies $\log_{2}(8) = 3$. The definition also leads to the identity $b^{\log_{b}(x)} = x$.

Natural logarithm (base $e$)

The natural logarithm uses the constant $e \approx 2.718281$ as its base. It is typically written as $\ln(x)$ or $\log_{e}(x)$. The number $e$ arises naturally in contexts where growth compounds continuously. Consider a bank deposit with an annual interest rate of 100% and very frequent compounding. The growth factor after one year is $\left(1 + \frac{1}{m}\right)^{m}$, where $m$ is the number of compounding periods per year. As $m$ increases, this factor approaches $e$:

$m$	$\left(1 + \frac{1}{m}\right)^{m}$
1	2
10	2.59374
100	2.70481
1,000	2.71692
10,000	2.71814
100,000	2.71826
1,000,000	2.71828

Thus, $e \approx 2.71828$. Natural logarithms appear frequently in calculus, finance (e.g., continuously compounded interest), economics (e.g., GDP growth rates), and the natural sciences.

Common logarithm (base 10)

The common logarithm, denoted $\log_{10}(x)$ or $\lg(x)$, uses 10 as its base. It is also called the decimal logarithm. Before electronic calculators, common logarithm tables were widely used for hand computations. Henry Briggs, an English mathematician, popularized this base in the 17th century. Below are some common logarithm values for both bases:

$x$	$\log_{10}(x)$	$\ln(x)$
1	0	0
2	0.30103	0.693147
3	0.477121	1.098612
4	0.60206	1.386294
5	0.69897	1.609438
10	1	2.302585
100	2	4.60517
1,000	3	6.907755

Calculating with an arbitrary base

If you need the logarithm of $x$ with base $b$ and the calculator does not directly offer that base, you can apply the change‑of‑base formula:

$$\log_{b}(x) = \frac{\ln(x)}{\ln(b)} = \frac{\log_{10}(x)}{\log_{10}(b)}.$$

Our log calculator automatically applies this formula, so you can input any positive base (other than 1) and get the correct result.

Example: log base 2

To compute $\log_{2}(100)$ manually using common logs:

1. Find $\log_{10}(100) = 2$.
2. Find $\log_{10}(2) = 0.30103$.
3. Divide: $2 / 0.30103 \approx 6.644$.

Thus, $\log_{2}(100) \approx 6.644$. With the calculator, you simply enter the number 100 and the base 2 to obtain the answer directly.

Key properties of logarithms

These fundamental rules apply for any valid base $b>0$, $b\neq1$ and are useful for simplifying expressions:

• Product rule: $\log_{b}(xy) = \log_{b}(x) + \log_{b}(y)$
• Quotient rule: $\log_{b}\!\left(\frac{x}{y}\right) = \log_{b}(x) - \log_{b}(y)$
• Power rule: $\log_{b}(x^{y}) = y \log_{b}(x)$
• Log of the base: $\log_{b}(b) = 1$
• Log of 1: $\log_{b}(1) = 0$
• Reciprocal rule: $\log_{b}\!\left(\frac{1}{x}\right) = -\log_{b}(x)$

A brief history of logarithms

The concept of logarithms can be traced back to 8th‑century India, but the modern idea was published by the Scottish mathematician John Napier in 1614 in his work Mirifici Logarithmorum Canonis Descriptio. His goal was to simplify the tedious astronomical calculations of the time. Henry Briggs visited Napier and together they developed the first table of base‑10 logarithms, published in 1624. Later, the slide rule—a physical device that used logarithmic scales—was invented by William Oughtred around 1622 and became an essential tool for engineers, scientists, and even astronauts until the electronic calculator era.

Real‑world applications of logarithms

Logarithms describe many natural phenomena. A logarithmic spiral appears in the shells of nautiluses, in galaxies, and in cyclones. Sound intensity is measured in decibels (dB), which are logarithmic. The Richter scale for earthquakes, the pH scale for acidity, and the Mohs scale for mineral hardness are all logarithmic. In finance, the natural logarithm is used to model continuously compounded interest and to compute growth rates. In science, logarithms help in analyzing exponential growth and decay, such as population growth and radioactive decay.

Using logarithms to simplify arithmetic

Before pocket calculators were common, logarithms were a powerful tool for multiplication, division, exponentiation, and root extraction. The product rule $\log_{b}(xy) = \log_{b}(x) + \log_{b}(y)$ allowed multiplication to be performed as addition. As an example, to find $5.89 \times 4.73$ without a calculator:

1. Look up $\log_{10}(5.89) \approx 0.7701153$ and $\log_{10}(4.73) \approx 0.674861$.
2. Add them: $0.7701153 + 0.674861 = 1.4449763$.
3. Find the antilog (10 raised to that power) to get $10^{1.4449763} = 10^{0.4449763} \times 10^{1}$. The antilog of $0.4449763$ is about $2.78597$, so the product is approximately $27.8597$.

This method also handles massive numbers, such as $100!$. By summing the logs of the integers from 1 to 100, we obtain $\log_{10}(100!) \approx 157.97$, so $100! \approx 10^{0.97} \times 10^{157} = 9.33 \times 10^{157}$.

The free log calculator online makes such calculations trivial, but understanding the underlying principle reveals why logarithms are so valuable.