😍 Log calculator (logarithm calculator)

In mathematics and other exact sciences, logarithms are widely used - functions that are the inverse of raising to a power. For example, the logarithm of 10 of 1000 is equal to 3, since to obtain a thousand, 10 must be cubed, and the logarithm of 2 of 16 is equal to 4, since 16 is 2 to the fourth power.

Logarithms greatly simplify complex mathematical calculations, since they can be used to express exponentiation and root extraction as multiplication and division by the exponent.

In addition to logarithms, their inverse functions are also used in the exact sciences - antilogarithms, or “inverse logarithms”. The antilogarithm of x is the result of potentiation, or a number whose logarithm is equal to x.

Logarithms are denoted in formulas as log, and antilogarithms are denoted as ant log. These designations can be found not only in logarithmic tables, but also on the keyboards of engineering calculators. But today, to calculate these functions, special online calculators are more often used - much more convenient and accessible.

The history of logarithms

Although the logarithmic function was invented much later, the prerequisites for its appearance were traced back to Antiquity. For example, the ancient Greek scientist Archimedes in the 3rd century BC established a connection between arithmetic and geometric progression, and investigated the properties of powers with natural exponents.

But tables of integer exponents (for bases 2, 3 and 4), which can be called logarithms in their modern sense, were obtained only in the 8th century - by the Indian scientist Virasena.

As astronomy and navigation developed, an increasingly urgent need arose to simplify complex mathematical calculations: multiplying and dividing multi-digit numbers, extracting roots, raising to powers.

In 1544, the German scientist Michael Stiefel took a decisive step in this direction by compiling a logarithmic table that was later named after him. The idea of comparing arithmetic and geometric progressions using tables was described by Stiefel in the book Arithmetica integra, and formed the basis for subsequent works by Nikolai Oresme and Nicola Chuquet.

In addition to them, the Scottish mathematician John Napier studied logarithms, who published Mirifici Logarithmorum Canonis Descriptio in Latin in 1614. This work described not only the properties of the logarithmic function, but also eight-digit tables of logarithms of sines, cosines and tangents. According to one version, it was Knepper who approved the name “logarithm”, which since the 17th century has become the only one and has no alternative.

Despite his serious contributions to science, John Knepper made a number of errors when compiling logarithmic tables (for numbers after the sixth digit), and they were disputed in 1620-1624.

In 1624, Johannes Kepler published his version of the logarithmic tables Chilias logarithmorum ad totidem numeros rotundos, and Edmund Wingate and William Oughtred invented the first slide rule. The British scientist Henry Briggs also carried out parallel research, and in 1617 he compiled 14-digit tables of decimal logarithms.

As in the case of Knepper's work, errors were also subsequently discovered in Briggs' tables. Initially, the table described decimal logarithms from 1 to 1000 with 8 decimal places, but after recalculation their number increased to 14 decimal places. In 1783, Georg Vega published a revised version, and on its basis the Bremiker tables were compiled - absolutely accurate and error-free.

It was the ready-made tables of logarithms that made this mathematical function so widespread and in demand. After all, now, instead of complex calculations, it was enough to check the required column and instantly get the desired result. The French mathematician Pierre-Simon Laplace said that the invention of logarithms “by shortening the astronomer’s work, doubled his life.”

In the 19th century, logarithms began to be used in complex analysis. In particular, Carl Friedrich Gauss in 1811 developed a theory of the multivaluedness of the logarithmic function, defined as the integral of 1/z. And Georg Friedrich Bernhard Riemann built a general theory of Riemann surfaces based on logarithms.

Today logarithms are used in algebra, geometry, physics, astronomy, engineering, economics and many other sciences. If earlier these functions were calculated manually or using logarithmic tables, and at the turn of the 20th-21st centuries - with the help of engineering calculators, today computer technology is used for this purpose. Just run the appropriate online calculator, and it will calculate the logarithm in a split second!

Today, the logarithmic function is used wherever there are processes that change over time. Biology, chemistry, physics, electrical engineering - all these sciences use logarithms. But the logarithm is most widespread in economics - as an integral part of mathematical analysis.

Definition

According to mathematical definition, the logarithm of a number x is an exponent to which the base b must be raised to obtain x. In other words, a logarithmic function is an inverted power, or the inverse of a power.

For example, the expression 2^3 = 8 can be represented as: log2(8) = 3. 2 is the base of the power, 3 is the exponent, 8 is the result of raising to the power. For clarity, we can list other examples:

log3(9) = 2.
log7(343) = 3.
log2(0.25) = −2.
log625(125) = 0.75.

In general, the expression loga(b) is read as “the logarithm of b to base a.” When writing a logarithm, it is important not to confuse the base with the power. Thus, in the expression log2(8) = 3, 2 is the base and 3 is the power.

Varieties of logarithms

There are three main types of logarithms: natural, decimal and binary.

Natural

When working with irrational numbers (which cannot be written as a common fraction with a numerator and denominator), use the natural logarithm, the base of which is the Euler number (e) equal to 2.71828. The standard expression of the form loge(x) in this case is replaced by ln x. ln is the natural logarithm, the designation of which can be found on most engineering calculators.

Why do you need to combine the logarithm with the Euler number? This is necessary to carry out exponential calculations. Thus, a whole series of quantities changes over time not proportionally, but exponentially - with a changing speed.

Using the exponent, you can describe the process of bacterial reproduction, the growth of snowballs, an increase in biological populations, an increase in income, and the decay of radioactive substances. The natural logarithm allows us to describe these phenomena. For example, crystal growth can be expressed this way:

e^3 = 20.086.
e^5 = 148.413.
e^10 = 22026.466.

Here e is Euler's constant number, and the degree is the number of years. That is, in 3 years the crystal will grow to 20.086 kilograms, in 5 years - to 148.413 kilograms, and so on. Similar calculations can be carried out for any other exponential quantities.

Decimal and binary

These varieties were introduced into circulation to simplify the writing of formulas and equations in decimal and binary systems. Thus, the decimal logarithm always has a base of 10. It is denoted as log x and corresponds to the expression log10(x). In turn, the binary logarithm always has a base of 2 and is denoted as lb x. It can also be represented as log2(x).

Properties and calculation formulas

When working with logarithms, you can use simplification and transformation formulas that are valid for any values of the variables. The most popular formulas include the following:

loga(1) = 0 - the logarithm of one is always zero.
loga(a) = 1 - if the number and base coincide, the logarithm is always equal to one.
loga(a^n) = n is the basic logarithmic identity.
loga(bc) = loga(b) + loga(c) - the logarithm of the product of numbers is equal to the sum of their logarithms.
loga(b/c) = loga(b) − loga(c) - the logarithm of a fraction is equal to the difference between the logarithms of the numerator and denominator.
loga(b^n) = n × loga(b) - the degree of bases and arguments can be extended beyond logarithms.

Knowing these transformation rules, you can simplify and find the values of logarithms in specific problems. For example, let's calculate the expression lg2 × lb10. After transformation we get: log10(2) × log2(10). After reduction to a single base we get:

1/log2(10) × log2(10) = log2(10)/log2(10) = 1.

Consider another example with the addition of logarithms: log216(2) + log216(3). Using the transformation formulas, we get:

log216(2) + log216(3) = log216(2 × 3) = log216(6) = log6^3(6) = (1/3) × log6(6) = (1/3) × 1 = 1/3.

Finally, the simple expression log3(81) can be written as log3(9 × 9) and calculated as log3(3^2+2) = log3(3^4) = 4 × log3(3) = 4 × 1 = 4. That is, there is a high probability that after the reductions there will not even be any logarithmic functions left in the expression, and it will be presented as a number that has already been calculated.

Using conversion formulas, you can shorten expressions and reduce them to a simpler form of notation, and only then carry out their calculation. If the intermediate values of the transformations are not important to you, but only the final result, you should use our online calculator. You just need to enter the required values into it, and the program will calculate the decimal, natural or binary logarithm in a fraction of a second!