Logarithmic Function

In mathematics, prior to the development of calculus, many mathematicians relied on logarithms to convert multiplication and division problems into addition and subtraction problems. This was done before calculus was discovered. To arrive at a different number using logarithms, the power is multiplied by some numbers (often, the base number). It is the complement of a function known as the exponential function. It is common knowledge that the fields of mathematics and science frequently deal with high powers of numbers; hence, logarithms are quite significant and helpful. 

Logarithmic functions definition: 

In mathematics, the exponentiation function is analogous to the logarithmic function, which is an inverse function. The definition of the logarithmic function is as follows: 

For x > 0 , a > 0, and a≠1,

y= log_a x if and only if x = a^y

Then the function is given by

f(x) = log_a x 

The letter a serves as the basis of the logarithm. One way to interpret this is as the log base an of x. Base 10 and base e are the two bases that are utilised in logarithmic functions the majority of the time. 

History of logarithms: 

In Europe during the seventeenth century, a new function was found that extended the realm of analysis beyond the scope of algebraic methods. This discovery is considered to be the beginning of the history of logarithms. In 1614, John Napier published a treatise titled Mirifici Logarithmorum Canonis Descriptio, in which he advocated openly for the use of logarithms as a mathematical approach (Description of the Wonderful Rule of Logarithms). Before Napier’s invention, there had been other methods with similar scopes, such as the prosthaphaeresis or the use of tables of progressions, both of which were extensively developed by Jost Bürgi around the year 1600. These methods existed prior to Napier’s invention. The word “logarithm” originates from Middle Latin, and it was coined by Napier. Its literal translation from Greek is “ratio-number,” which comes from the words logos (meaning “proportion, ratio, word”) and arithmos (meaning “number”). 

Common logarithmic functions: 

The logarithmic function that uses 10 as its base is referred to as the common logarithmic function, and its notation is either log₁₀ or just log. 

f(x) = log₁₀ x 

Natural logarithmic functions: 

The logarithmic function with e as the base is referred to as the natural logarithmic function, and the symbol for this function is log_e. 

f(x) = log_e x 

Logarithmic function properties: 

When the input is in the form of a product, quotient, or the value taken to the power, logarithmic functions have certain qualities that enable you to simplify the logarithms. The following is a list of some of the available properties: 

Product rule: 

log_b MN = log_b M + log_b N 

After performing this operation, add the exponents to the product of the two numbers that have the same base. 

Example : log 30 + log 2 = log 60 

Quotient rule: 

log_b M/N = log_b M – log_b N 

After dividing the two integers, subtract the exponents to get the final answer. 

Example : log₈ 56 – log₈ 7 = log₈(56/7) = log₈8 = 1 

Power rule: 

You can multiply the exponents of an exponential expression by raising it to a higher power. 

Log_b Mp = P log_b M 

Example : log 100³ = 3. Log 100 = 3 x 2 = 6 

Zero exponent rule: 

Log_a 1 = 0.

Change of base rule: 

log_b (x) = ln x / ln b or log_b (x) = log₁₀ x / log₁₀ b 

In addition to that, there are a few logarithmic functions that work with fractions. It is possible to determine the logarithm of a fraction by applying the identities to it, which is a useful quality. 

• ln(ab)= ln(a)+ln(b)

• ln(a^x) = x ln (a) 

Applications of logarithmic functions: 

Logarithmic scales condense a broad range of values down into a more manageable scope. The decibel, or dB, is one example of a unit that can be used to express ratios as logarithms. This is typically done for signal power and amplitude (of which sound pressure is a common example). In chemistry, the logarithmic measurement of acidity in aqueous solutions is referred to as the pH scale. Logarithms are frequently used in mathematical formulas, as well as in the evaluation of the complexity of algorithmic structures and fractals, which are types of geometric objects. They contribute to the description of frequency ratios of musical intervals, appear in formulas counting prime numbers or approximating factorials, inform some models used in psychophysics, and can be of use in forensic accounting. 

Solved examples: 

Q1. Use the properties of logarithms to write as a single logarithm for the given equation: 5 log₉ x + 7 log₉ y – 3 log₉ z 

Ans. By using the power rule , Log_b M^p = P log_b M, we can write the given equation as

5 log₉ x + 7 log₉ y – 3 log₉ z = log₉ x⁵ + log₉ y⁷ – log₉ z³

From product rule, log_b MN = log_b M + log_b N

5 log₉ x + 7 log₉ y – 3 log₉ z = log₉ x⁵y⁷ – log₉ z³

From Quotient rule, log_b M/N = log_b M – log_b N

5 log₉ x + 7 log₉ y – 3 log₉ z = log9 (x⁵y⁷ / z³ )

Therefore, the single logarithm is 5 log₉ x + 7 log₉ y – 3 log₉ z = log₉ (x⁵y⁷ / z³ ) 

Conclusion: 

The idea that the logarithm is the inverse of the exponentiation process can be applied to a variety of other mathematical structures as well. The logarithm, on the other hand, is typically a function that can take on several values in most contexts. For instance, the multi-valued inverse of the complex exponential function is the complex logarithm. In a similar vein, the discrete logarithm is the multi-valued inverse of the exponential function in finite groups. This function has applications in public-key cryptography.