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 » 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
x²- 2x = 15
x²- 2x - 15 = 0
x²+ 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}.