# CLASS-11LOGARITHMIC FUNCTION

LOGARITHMIC FUNCTION -

The logarithmic function, often denoted as logb​(x), represents the exponent to which a given base b must be raised to obtain a given number x. In other words, it is the inverse operation of exponentiation. The most common bases for logarithmic functions are e (natural logarithm, denoted as ln⁡()ln(x)) and 10 (common logarithm, denoted as log⁡()log(x)).

1. Common Logarithm (log⁡()log(x)): This is the logarithm function with base 10. It is defined as:

log⁡()=log⁡10()log(x)=log10​(x)

For example, log⁡(1000)=3log(1000)=3 because 103=1000103=1000.

1. Natural Logarithm (ln⁡()ln(x)): This is the logarithm function with base e, where e is Euler's number, approximately equal to 2.71828. It is defined as:

ln⁡()=log⁡()ln(x)=loge​(x)

For example, ln⁡()=1ln(e)=1 because 1=e1=e.

Key properties of logarithmic functions include:

• Inverse Property: The logarithmic function is the inverse of the exponential function. This means that log⁡()=logb​(bx)=x and log⁡()=blogb​(x)=x for all appropriate values of x and b.
• Product Property: log⁡()=log⁡()+log⁡()logb​(xy)=logb​(x)+logb​(y). This property allows you to break down logarithms of products into sums of logarithms.
• Quotient Property: log⁡()=log⁡()−log⁡()logb​(yx​)=logb​(x)−logb​(y). This property allows you to break down logarithms of quotients into differences of logarithms.
• Power Property: log⁡()=⋅log⁡()logb​(xp)=p⋅logb​(x). This property allows you to bring exponents down in front of logarithms.