Antilog Calculator

Calculate the antilog_bX value:

In mathematics, the logarithm of a number is the power that the base must be raised to, to obtain this number. For example, if ‘x’ is the number obtained when the base ‘b’ is raised to the power ‘y’, then the log operation will yield the number ‘y’, which is the power that the base ‘b’ must be raised to, in order to obtain ‘x’. The exponentiation is written as . The log notation is . This is read as ‘the log of x to the base b’. This basically shows that the number base ‘b’ must be raised to the power ‘y’ to obtain the value ‘x’.

The logarithm function is the inverse of exponentiation. This essentially means that antilog is simply the exponentiation. To find the antilog of a number ‘y’, the base ‘b’ needs to be raised to the power of ‘y’. Therefore, x = log_bx^-1(y)=b.

For example, 3=log₂8. The antilog is 8=2³. The log can be calculated for only positive real numbers, and the base is a positive number not equal to 1.

To find the antilog of a number, we start with the base. Commonly, the base 10 is chosen. Next, we need the number we need to find the antilog of, say 3. To find the antilog, simply raise the base to this number. Hence, 10³=1000. If the base e is chosen, then we find the natural antilog, which is e³=20.086

Given below are some important properties of antilog.

Base – Denoted by a subscript, the base of a log ‘b’ is the number which needs to be multiplied with itself a certain number of times to get the number ‘x’. When the base is Euler’s number, the log is called natural log. To find the antilog, the base is raised to this number.

Number (or exponent) – The chosen number (or exponent) is the number to which the base is being raised to obtain the antilog. For example, for b^x, we say that the number ‘b’ is being raised to the power ‘x’.

Antilog - This is the number obtained when the base is raised to a given number. In the formula, y = b^x, ‘y’ denotes the antilog. The antilog is the same number which is obtained after exponentiation.

The formula for log is y = log_b x , which is equivalent to x = b^y in exponent form.

The formula for antilog is x = log_b x^-1(y) = b^y.

Log and antilog are inverse functions, so x = b^y = b^x and x = (b^y)

For natural logarithm, the formula for log is y = log_ex = ln x, which is equivalent to x = e^y in exponent form. The formula for antilog is x = log_e x^-1(y) = ln x^-1(y) = e^y.

Natural log and natural antilog are inverse functions, so x = e^y = b^{ln x} and y = ln x = ln(e^y)

Moreover, log_b(antilog_by) = y and antilog_b (log_by) = y

For the natural antilog, log_e(antilog_ey) = y and anitlog_e(log_ey) = y

The log and antilog can be found directly using log and antilog tables. To find the antilog of a number, we need to know the parts of it. The characteristic part of a number is the part before the decimal, and the mantissa is only the part after the decimal point.

Consider the number 2.5417.

2 is the characteristic part and 0.5417 is the mantissa.

Now find the corresponding value of the mantissa using the antilog table. Find the row number which is equal to .54 and the choose the column number as 1. Note down the corresponding value, it is 3475.

Now look at the mean difference column. For this step also, use the .54 row and choose the column number as 7. The corresponding value is 6.

Add the values obtained in the previous two steps, 3475 + 6 = 3481.

Now we include the decimal point. The characteristic part was 2. Add 1 to this, to get 3. So, there will be 3 numbers before the decimal point.

Hence, the antilog of 2.5417 is 348.1

Log and antilog are frequently used when dealing with large numbers, and have applications in various fields of science and technology. Some examples are:

In chemistry, log and antilog are used in the pH scale to measure the acidity or basicity of a substance. It is defined as pH = −log[H⁺] , pOH = −log[OH^-] and pKw = −log[Kw] . This range varies from 0 to 14, with 7 being neutral. The pH of water is 7. A substance having pH less than 7 is acidic, and it is basic when the pH is greater than 7.
For example, if an acid has an H⁺ concentration of 0.001M, to find the pH we first need to convert the number to exponential form, find the log and then ultimately solve the pH equation. H⁺ = 0.001M = 10^-3
log 10^-3 = −3
pH = −log[H⁺] = −log[10^-3] = −(−3) = +3
The negative sign in the log definition is used to obtain a positive pH value.

In acoustics, the sound pressure and sound intensity, measured in Decibels, make use of log. The formula is
Sound pressure = 20 log (P / P₀)
where the sound pressure is measured in dB, P=pressure of the sound wave in pascals, and P₀ is the reference value of sound pressure.
Similarly, the sound intensity formula is
Sound intensity = 20 log (I / I₀) where the sound intensity is measured in dB, I = sound intensity in W/m², and I₀ is the reference value of sound intensity. The range of sound can go from extremely quiet, like a rustling leaf (10 dB) to extremely loud, like a supersonic aircraft (140 dB).

The Richter scale, which measures the intensity of earthquakes on a single scale with a small range (usually from 1 to 10), also makes use of log and antilog. The increase of each point shows that the intensity of the earthquake is increasing.

Some other applications can be found in the fields of probability and statistics, and number theory.

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