*e* and the Natural Logarithm

One of the most important numbers used as the base for exponential and logarithmic functions is denoted as *e*. It is an irrational number, with its value approximated as *e* Â» 2.7182818. Exponential and logarithmic functions with base *e* occur in many practical applications, including those involving growth and decay, continuous compounding of interest, alternating currents and learning curves.

The natural logarithm of a positive real number is defined as the logarithm to the base *e* of the number. The natural logarithm of x, x > 0, is denoted as ln x. Symbolically,

ln x = log

_{e}x where x>0

By definition, ln x = y implies that e^{y} = x.

When converting from base 10 to base *e*, we can use the following formula:

log x = .4343 ln x

where log _{10} e = .4343.

Since the function f(x) = e^{x} and f(x) = ln x are inverse functions of each other,

ln e^{x} = x and e^{ln x }= x

The natural logarithm possesses the same properties as common logarithms.

EX.

e^{x + 3 }= 17

ln e^{x + 3} = ln 17

x + 3 = ln 17

x + 3 = 2.833

x = -0.167

EX.

ln x + ln (x - 2) = ln 15

ln [x(x - 2)] = ln 15

x^{2}- 2x = 15

x^{2}- 2x - 15 = 0

x^{2}+ 3x -5x - 15 = 0

x(x + 3) -5(x + 3) = 0

(x - 5)(x + 3) = 0

x - 5 = 0 or x + 3 = 0

x = 5 or x = -3

Since x cannot be negative, the solution set is {5}.