Logarithm

A logarithm is a mathematical operation that represents the inverse of exponentiation. If b^x = y, then log_b (y) =x

Here, b is the base, y is the result of raising the base to the power of x, and x is the logarithm.

log_by is the power by which b must be raised to obtain y.

1.0Logarithm Definition

A logarithm is a mathematical function that yields the exponent to which a fixed number, termed the base, must be raised to obtain a specified number. In other words, if you have a logarithm of a number "y" with base "b", it is denoted as “log_b(y)” and it represents the power to which the base "b" must be raised to yield the value "y".

For example, in base 10 logarithms, if you have log₁₀(1000) = 3, it means 10 raised to the power of 3 equals 1000 (10³= 1000).

Logarithms are widely used in various fields, such as mathematics, engineering, science, economics, and computer science for simplifying calculations involving large numbers, expressing exponential growth or decay, solving equations, and more. 

Note: For log_by to be defined, y > 0, b > 0, b ≠ 1.

There are several common bases for logarithms

• Natural logarithm (base e)

Denoted as ln y, where the base e is called natural logarithm, (e approximately equal to 2.71828). The natural logarithm function is defined for positive real numbers y, and it represents the power to which e must be raised to obtain y.

Mathematically, the natural logarithm function satisfies the equation: ln y = x

If and only if: e^x = y

The natural logarithm function is widely used in various fields of mathematics, including calculus, differential equations, probability, and statistics, as well as in sciences such as physics, chemistry, and biology. It has numerous applications in modeling exponential growth and decay, solving equations involving exponential functions, calculating compound interest, and more.

• Base 10 logarithm:

Denoted as log₁₀ (y), is a logarithm with base 10. It represents the power to which 10 must be raised to obtain y.

Mathematically, the base 10 logarithm function satisfies the equation: log₁₀ (y)=x

If and only if: 10^x = y

The base 10 logarithm is commonly used in various fields, especially in science, engineering, and technology. It is often used for practical purposes because of its easy conversion to and from the common logarithm. For example, the pH scale in chemistry and the Richter scale in seismology are both based on base 10 logarithms. 

• Other bases:

Logarithms can be taken with any positive base other than 1, though natural logarithms (base e) and base 10 logarithms are most common.

2.0Logarithm Graph

3.0Logarithm Properties

1. Product Rule:

log_b (xy)=log_b (x)+log_b (y)

This property states that the sum of the logarithms of the individual factors is equal to the logarithm of a product. 

2. Quotient Rule:

${\log }_b\left(\frac{x}{y}\right)={\log }_b(x)-{\log }_b(y)$

This property states that the logarithm of the quotient is equal to the difference of the logarithms of the numerator and denominator.

3. Power Rule:

log_b (x^a) = a. log_b (x)

This property states that the logarithm of a power is equal to the exponent times the logarithm of the base.

4. Change of Base Formula:

${\log }_{\mathrm{b}}(\mathrm{x})=\frac{{\log }_{\mathrm{c}}(\mathrm{x})}{{\log }_{\mathrm{c}}(\mathrm{b})}$

This formula allows us to change the base of a logarithm

5. Inverse Property:

log_b (b^x) = x

This property states that applying a logarithm with base b to a number raised to the same base b yields the original exponent.

6. Zero Property:

log_b (1) = 0

The logarithm of 1 w.r.t any base other than 1 remains constant at zero.

7. Negative Property:

${\log }_b\left(\frac{1}{x}\right)=-{\log }_b(x)$

This property states that the logarithm of the reciprocal of a number equals the negative of the logarithm of the number itself.

Some More Important Properties of Logarithm

1. ${\mathrm{a}}^{{\log }^{\mathrm{b}}}=\mathrm{b}$
2. $a^{\log c^b}=b^{\log c^a}$

provided log are defined.

4.0Logarithm Solved Questions

1. Evaluate: – log₂(256)

⇒   2⁸ = 256

⇒  log₂(256) = 8

2. Simplify    log₃(81) – log₃ (9)

=  ${\log }_3\left(\frac{81}{9}\right)$ (by Division Rule)

=   log₃ (9)

= 2  (3² =9)

3. Find the value of  $2\log \left(\frac{2}{5}\right)+3\log \left(\frac{25}{8}\right)+\log \left(\frac{128}{625}\right)$

= $\log \frac{2^2}{5^2}+\log {\left(\frac{5^2}{2^3}\right)}^3+\log \frac{2^7}{5^4}$

= $\log \frac{2^2}{5^2}\cdot \frac{5^6}{2^9}\cdot \frac{2^7}{5^4}=\log 1=0$

4. log₃ (x) = log₃ (8) + log₃ (5)

= log ₃ (8 × 5)

= log₃ (40)

thus, x = 40