Exponentials and Natural Logarithms

Exponentials and Natural Logarithms

Natural Exponentials

Natural exponential functions have the form y = e^x, where e is the natural base approximately equal to 2.71828.
It’s vital to understand e as the base rate of growth shared by all continually growing processes.
Much like other exponentials, the natural exponential function is always positive: e^x > 0 for all real x.
These functions possess a horizontal asymptote at y=0.

Natural Logarithms

The natural logarithm, denoted by ln, is the logarithm to the base e. It’s often written in the form y = ln(x).
Natural logarithms transform multiplicative processes into additive ones: ln(a.b) = ln(a) + ln(b) and ln(a/b) = ln(a) - ln(b).
The inverse property for natural logs and exponentials is ln(e^x) = x and e^ln(x) = x.
For all real x > 0, the derivative of ln(x) is 1/x.

Laws of Natural Logarithms and Exponentials

Remember important laws of logarithms when dealing with natural log functions: ln(aⁿ) = n.ln(a), ln(1) = 0, and ln(e) = 1.
For the natural exponential, remember e^x+y = e^x.e^y and (e^x)^y = e^x.y.

Solving Natural Exponential and Logarithm Problems

To solve natural exponential problems like e^x = a, rewrite them in natural logarithm form: x = ln(a).
To solve natural logarithmic problems like ln(x) = a, rewrite them in exponential form: e^a = x.
Always double-check your work. For instance, ln(x) doesn’t exist for x ≤ 0, so meticulously verify the domain of your answers.

Applications of Natural Exponentials and Logarithms

Natural exponentials are commonly encountered when analysing continuous growth or decay problems, such as those found in financial mathematics or scientific modelling.
Likewise, natural logarithms are used where exponential growth needs to be transformed into a more manageable linear form.
Natural logarithms are crucial in calculus, specifically when integrating the reciprocal function 1/x to get **ln x + c**.