CLASS-11
LOGARITHMIC 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)).
- Common Logarithm (log()log(x)): This is the logarithm function with base 10. It is defined as:
log()=log10()log(x)=log10(x)
For example, log(1000)=3log(1000)=3 because 103=1000103=1000.
- 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.