Inverse Logarithm (Antilog) Calculator

Antilog calculator commonly called as Anti Log or Inverse Log Calculator is an online math calculator that calculates the inverse log value for the real number with respect to the given or natural base values. Using this calculator, we will understand methods of how to find the antilogarithm of any number with respect to the given base.

It is necessary to follow the next steps:

1. Enter the logarithm value of number and the base of logarithm. The base of logarithm must be positive real number not equal to $1$.
2. Press the "CALCULATE" button to make the computation;
3. Antilog calculator will give the antilogarithm of real number number with the positive base not equal to 1.

Input: Two real numbers. The second one which represents the base of log must be positive real number not equal to 1;
Output: A positive real number.

Antilogarithm Rule: For some real numbers $a,b$ and $x$, if $\log_a x = b$ then $$x= {\rm antilog}_a b=a^b$$ where $a\in R^+\setminus\{1\}$

What is Logarithm?

To recall, exponential functions are very interesting as models for description of natural processes, physical magnitudes, as well as economic and social problems. A function $f(x)=a^x$, where $a>0$ is called an exponential function. The positive constant $a$ is called the base of the exponential function.
The most commonly encountered base of exponential function is the number $e=2.7182818...$ This number is called Euler's number. The exponential function with the base $e$ is $f(x)=e^x$ and it is often called the natural exponential function. The exponential function $f(x)=a^x$, for $a>0, a\ne 1$, is bijection so it has an inverse function. The inverse function $g(x)$ of $f(x)$ is $$g(x)=\log_a x$$ where $a>0, a\ne 1$. From the definition, it holds that $f(x)=\log_a x$ if and only if $x=a^{f(x)}$. This inverse function of the exponential function is called the logarithmic function for the base $a$.

What is and How to Find Inverse Logarithm?

In some problems, the logarithm of $x$ and the base $a$ are known, but $x$ is unknown. An antilogarithm is the inverse function of a logarithm. Since the base of an exponential function cannot be negative, the base of antilog is always a positive real number. Because the inverse of a logarithmic function is an exponential function, then $${\rm antilog}_a ( \log_a(x) ) = x$$ If $\log_a x = b$, then $x$ is called the antilogarithm of $b$ and is written as $$x= {\rm antilog}_a b=a^b$$ The antilogarithm in base $a$ of $b$ is therefore $a^b$. If the base of antilog is not written, ${\rm antilog}b$ is $10^b$, because $\log x$ means logarithm to the base $10$.
The next problem naturally arises: Find $x$, if we know the natural logarithm of a number $x$. The antilogarithm of a natural logarithm is written ${\rm antiln}\; x$. If $\ln x = b$, then $x ={\rm antiln}\; b$. Using the exponential function, we can find the antilogarithm of a natural logarithm.
As we can find the values of the logarithm from logarithm tables, we there exist antilogarithm tables that enable us to find the numbers whose logarithms are known.
Inverse log calculator shows the calculation for finding the antilogarithm in base $2$ of $10$. For any other combinations the base and logarithm, just supply the other two numbers as inputs and click on the on the "CALCULATE" button. Have in mind that the value of the base must be positive, not equal to $1$. The grade school students may use this Antilog calculator to generate the work, check an exponent power concept, verify the results or do their homework problems efficiently.

Real World Problems Using Antilogs

When we have an exponential function, immediately we can find the corresponding a logarithmic function. Since the antilogarithmic function is the exponential function, the applications of antilogarithmic function are actually applications of the exponential function. The exponential functions are so useful in real-world situations. For example, they are used to model population growth, exponential decay, and compound interest.

Inverse Log Practice Problems

Practice Problem 1 :
Solve equation in $x$, $9^{3x}=54$.

Practice Problem 2 : Meteorologists determined that for altitudes up to $10$ kilometers, the pressure $p$ in millimeters of mercury, is $p = 600e^{-0.112a}$, where $a$ is the altitude in kilometers. Find is the atmospheric pressure at the altitude of $5$ kilometers.

The Antilog calculator, formula, example calculation, real world problems and practice problems would be very useful for grade school students (K-12 education) to understand the concept of exponents and logarithm. This concept can be of significance in calculus, algebra, probability and many other fields of science and life.