Logarithmic Functions
The logarithmic function is an important medium of math calculations. Logarithms were discovered in the 16^{th} century by John Napier a Scottish mathematician, scientist, and astronomer. It has numerous applications in astronomical and scientific calculations involving huge numbers. Logarithmic functions are closely related to exponential functions and are often considered as an inverse of the exponential function. The exponential function a^{x} = N is transformed to a logarithmic function \(log_aN = x\).
The logarithm of any number X if interpreted as an exponential form, is the exponent to which the base of the logarithm should be raised, to obtain the number X. Here we shall aim at knowing more about logarithmic functions, type of logarithms, the graph of the logarithmic function, and the properties of logarithms.
What Are Logarithmic Functions?
The logarithmic function is derived from an exponential function. Some of the nonintegral exponent values can be calculated easily with the use of logarithmic functions. Finding the value of x in the exponential expressions 2^{x} = 8, 2^{x} = 16 is easy, but finding the value of x in 2^{x} = 10 is difficult. Here we can use logarithmic functions to transform 2^{x} = 10 into logarithmic form as \( log_210 = x\) and then find the value of x. The logarithm counts the numbers of occurrences of the base in repeated multiples. The formula for transforming an exponential function into a logarithmic function is as follows.
The exponential function of the form a^{x} = N can be transformed into a logarithmic function \( log_aN = x\). The logarithms are generally calculated with a base of 10, and the logarithmic value of any number can be found using a Napier logarithm table. The logarithms can be calculated for positive whole numbers, fractions, decimals, but cannot be calculated for negative values.
Graph of Logarithmic Functions
The domain of a logarithm function is similar to a square root function. The domain of the logarithmic function is plotted on the xaxis and the range is plotted with respect to the yaxis. Since the logarithm of 1 to any base (\(\log_n1 = 0\)) is equal to 0, the logarithmic function at the domain value of 1 cuts the xaxis. Also, the logarithm of zero does not exist, and hence the graph does not pass through the origin.
Natural Logarithms and Common Logarithms
The logarithms are broadly classified into two types, based on the base of the logarithms. We have natural logarithms and common logarithms. Natural logarithms are logarithms to the base 'e', and common logarithms are logarithms to the base of 10. Further logarithms can be calculated with reference to any base, but are often calculated for the base of either 'e' or '10'. The natural logarithms are written as \(\log_ex \), and the common logarithms are written as \(\log_{10}x\). To obtain the value of x from natural logarithms, it is equal to the power to which e has to be raised to obtain x, ie \(e^{1.6.9} = 5\).
e = 2.718
\(\log_eN = 2.303 ×\log_{10}N\)
\(\log_{10}N = 0.4343 × \log_eN\)
The value of e = 2.718281828459, but is often written in short as e = 2.718. Also, the above formulas help in the interconversion of natural logarithms and common logarithms.
Properties of Logarithmic Functions
Logarithmic function properties are helpful to work across complex logarithmic functions. All the general arithmetic operations across numbers are transformed into a different set of operations within logarithms. The product of two numbers, when taken within the logarithmic functions is equal to the sum of the logarithmic values of the two functions. Similarly, the operations of division are transformed into the difference of the logarithms of the two numbers. Let us list all the properties of logarithmic functions in the below points.
 logab = loga + logb
 loga/b = loga  logb
 \(log_b a = \frac{loga}{logb}\)
 loga^{x} = x loga
 \(log_1 a \) = 0
 \(log_a a \) = 1
Derivative of Logarithmic Functions
The derivation of the logarithmic function gives the slope of the tangent to the curve representing the logarithmic function. The functions within the logarithm can be a simple variable or a trigonometric ratio. The derivative taken for the logarithmic function gives the inverse of the logarithmic function as an answer. The formula for the derivative of the logarithmic function is as follows.
d/dx. Logx = 1/x
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Solved Examples on Logarithmic Functions

Example 1: Express 4^{3} = 64 in logarithmic form.
Solution: The exponential form a^{x} = N can be written in logarithmic function form as \(\log_aN = x \).
Hence, 4^{3 }= 64 can be written in logarithmic form as \(\log_4{64} = 3\)

Example 2: Simplify \(\log _{2} \frac{1}{128}\).
Solution: The simplification for the given logarithmic function expression is as follows.
\(\log _{2} \frac{1}{128}\) = \(\log_2 1  \log_2 128\)
= 0  \(\log_2 2^7\)
= \(\log_2 2^7\)
= 7 \(\log_2 2\)
= 7 (1)
=7
Answer: Hence \(\log _{2} \frac{1}{128}\) = 7
FAQs on Logarithmic Functions
How to Solve Logarithmic Functions?
The logarithmic function can be solved using the logarithmic formulas. The product of functions within logarithms is equal (logab = loga + logb) to the sum of two logarithm functions. The division of two logarithm functions(loga/b = loga  logb) is changed to the difference of logarithm functions. The logarithm functions can also be solved by changing it to exponential form.
How to Graph Logarithmic Functions?
The graph of logarithmic function y = log x can be obtained by plotting the logx value on the xaxis and the y value on the yaxis. The graph of logx does not pass through the origin, and the graph is similar to the square root graph.
How Are Exponential and Logarithmic Functions Related?
The exponential function of the form a^{x} = N can be transformed into a logarithmic function \( log_aN = x\). Here the exponential functions 2^{x} = 10 is transformed into logarithmic form as \( log_210 = x\), to find the value of x. The logarithm counts the numbers of occurrences of the base in repeated multiples.
How to Differentiate Logarithmic Functions?
The differentiation of a logarithmic function results in the inverse of the function. The differentiation of logx is equal to 1/x. (d/dx .logx = 1//x). Also, the antiderivative of 1/x gives back the logarithmic function.
What Is the Range of Logarithmic Functions?
The range of a logarithmic function takes all values, which include the positive and negative real number values. Thus the range of the logarithmic function is from negative infinity to positive infinity.
What Is the Domain of Logarithmic Functions?
The logarithms can be calculated for positive whole numbers, fractions, decimals, but cannot be calculated for negative values.. Hence the domain of the logarithmic function is a positive real number.
What Is the Formula for Logarithmic Functions?
The following formulas are helpful to work and solve the logarithmic functions.
 logab = loga + logb
 loga/b = loga  logb
 \(log_b a = \frac{loga}{logb}\)
 loga^{x} = x loga
What Are Logarithmic Functions Used For?
Logarithmic functions have numerous applications in physics, engineering, astronomy. The numeric measurements in astronomy include huge numbers with decimals and exponents. The huge scientific calculations can be easily simplified and calculated using logarithmic functions. The logarithmic functions help in transforming the product and division of numbers into sum and difference of numbers.