How Logarithmic and Exponential Function Related to Each Other

Functions are the foundation of calculus in mathematics. The functions are the many forms of relationships. In mathematics, a function is represented as a rule that produces a unique result for each input x. In mathematics, mapping is used to define a function. 

In mathematics, there infinite defined function are there, for example trigonometric function, logarithmic function, exponential function.

Exponential function

An Exponential function is a sort of mathematical function that contains exponents. A simple exponential function is defined as f(x) = b^x, where b > 0 and b ≠ 1.

Exponents can be used as exponential function, as the name state. However, an exponential function has a constant as its base and a variable as its exponent but not vice versa. An exponential function can have one of three forms.

• Domain of exponential Function

An exponential function’s domain is the set of all real numbers. 

• Range of exponential Function

The horizontal asymptote of the graph, say, y = d, and whether the graph is above or below y = d, may be used to determine the range of an exponential function.

logarithmic function

The logarithmic function is the inverse of exponentiation, In logarithms, the power of certain numbers (typically the base number) is raised to obtain another number.

• Range of Logarithmic Function

A logarithmic function’s range includes all values, including positive and negative real numbers. As a result, the logarithmic function has a range of negative infinity to positive infinity.

• Domain of Logarithmic Functions,

Logarithms can be computed for positive whole integers, fractions, and decimals, but not for negative values. As a result, the logarithmic function’s domain is the set of all positive real numbers.

In mathematics, an exponential function is a relationship of the type y = a^x, with the independent variable x spanning throughout the whole real number line as the exponent of a positive number a. The most significant of the exponential functions is y = e^x, sometimes written y = exp (x), in which e (2.7182818…) is the basis of the natural logarithm system (ln). Because x is a logarithm by definition, there is a logarithmic function that is the inverse of the exponential function. In particular, if y = e^x, then x = ln y.

Non-algebraic, or transcendental, functions are those that cannot be expressed as the product, sum, and difference of variables raised to a nonnegative integer power. The logarithmic and trigonometric functions are two more types of transcendental functions. Exponential functions are commonly used to explain and quantify a variety of events in physics, such as radioactive decay, in which the rate of change in a process or material is exactly proportional to its present value.

Relationship between logarithmic and exponential function

Logarithmic functions are closely connected to exponential functions and are regarded as the exponential function’s inverse.

Logarithmic functions are closely connected to exponential functions and are regarded as the exponential function’s inverse. The logarithmic function log_aN = x is created by transforming the exponential function a^x = N.

Comparison between logarithmic and exponential function

(x) = e^x denotes the exponential function, where e = lim(1 + 1/n)ⁿ = (2.718…) and is a transcendental irrational number. One of the function’s peculiarities is that its derivative is identical to itself; that is, when y = e^x, dy/dx = e^x. Furthermore, the function is an everywhere continuous rising function with an asymptote on the x-axis. As a result, the function is also one-to-one. We have that e^x> 0 for any x R, and it can be demonstrated that it is onto R⁺, Whereas,

The inverse of the exponential function is the logarithmic function. Because the exponential function is one-to-one and onto R⁺, a function g may be defined as a function from the set of positive real numbers into the set of real numbers provided by g(y) = x if and only if y=e^x. This function g is known as the logarithmic function or, more colloquially, the natural logarithm. It is represented as g(x) = log e^x = ln x.

Conclusion:

• If a^x = N then we can write log_aN = x

• The value of e is 2.7182818…