Logarithms

Logarithm- Introduction

Exponents can also be written as logarithms. We already know that 26=64. However, if we are asked to identify what number substitutes the question mark in 2x = 64, we can find the answer by trial and error: 6. But what if we’re asked to locate the question mark in equation 2? = 30? Is there a number that, when multiplied by two, equals 30? If not, what are the options for resolving the problem? Logarithms are the answer (or logs). To better understand the topic, go through this study material notes on logarithm thoroughly.

What is a logarithm?

Logs (or) logarithms are just another means of expressing exponents that can be utilized to address problems that can’t be solved just with exponents. It is not difficult to comprehend logs. It’s enough to know that a logarithmic equation is merely another method of writing an exponential equation to grasp logarithms. 

A logarithm is an inverse of a function to exponentiation which means the logarithm of a no. x is the exponent for which other no. is a fixed, the base b must be raised, producing the number x. The log calculates the no. of occurrences of the same factor in repeated products e.g. since 10000 = 10 x 10 × 10 × 10 = 10^4, the “logarithm base 10” of 10000 is 4, or log10 (10000) = 4. The logarithm of x to base b is denoted as log (x).

Types of Logarithm

The two types of the logarithm are as follows:

• Natural Logarithm

The base e logarithm is the natural logarithm. The natural logarithm is symbolized by the letters ln or loge. The Euler’s constant, approximately equal to 2.71828, is represented by “e.”

• Common Logarithm

The base ten logarithms are also known as the common logarithm. It is written as log10 or only log. The common logarithm of 1000, for example, is expressed as a log (1000). The common logarithm tells us how many times we must multiply 10 to reach the desired result.

Properties and Rules of Logarithm

Logarithmic operations can be performed on certain rules. The rules are further classified as:

• Multiplication/Product rule
• Derivative of log
• Division rule
• Change of base rule
• Exponential Rule/Power rule
• Base switch rule
• Integral of log

Multiplication/Product rule

The multiplication of 2 logarithmic values equals the sum of their logarithms.

Log(pq)= logbp + logbq

Derivative of log formula

If f (x) = logb (x), then the derivative of f(x) is given by,

f'(x) = 1/(x ln(b))

For example: Given, f (x) = log10 (x)

Then, f'(x) = 1/(x ln(10))

Division Rule

The division of 2 logarithmic numbers is equal to the difference of each logarithm.

Logb (p/q)= logbq – logbq

Change of Base Rule

Logb m = loga m/ loga b

Exponential Rule

In the exponential rule, the logarithm of m with a rational exponent is equal to the exponent times its logarithm.

Log (mn) = n logb m

Base Switch Rule

logb (a) = 1 / loga (b)

Integral of Log

∫logb(x)dx = x( logb(x) – 1/ln(b) ) + C

Some other standard properties

Other properties of logarithmic functions are:

• Logb b = 1
• Logb 0 = undefined
• Logb 1 = 0

Logarithmic Formulas

• logb(m/n) = logb(m) – logb(n)
• logb(mn) = logb(m) + logb(n)
• logbm√n =(1/m) logb(n)
• logb (xy) = y logb(x)
• logb(m+n) = logb m + logb(1+n/m)
• m logb(x) + n logb(y) = logb(xmyn)
• logb(m – n) = logb m + logb (1-n/m)

Conclusion

Logarithm properties are used to solve logarithm problems in mathematics. In elementary school, many algebraic properties were taught to us, such as commutative, associative, and distributive. Logarithmic functions have five basic characteristics. Using these properties, we may represent the log of a product as a sum of logs, the log of the quotient as a difference of logs, and the log of power as a product. The real number logs are seen in the positive number only. The logarithms of negative and complex numbers show complex logarithms.