Understanding Exponentials and Logs

Basic Concepts of Exponentials

An exponential function is a mathematical function of the form y = a^x, where ‘a’ is a positive real number, ‘x’ is the exponent, and ‘a’ is not equal to 1.
The number ‘a’ is usually referred to as the base of the exponential.
A special case of exponential functions is when the base is ‘e’, the base of natural logarithms, approximately equal to 2.71828182845904. This base is widely used in mathematics, physics, and engineering.
Exponential functions have the property that the rate of change (growth or decay) is directly proportional to the current value; that means the function grows or decays at a rate that increases or decreases in direct proportion with the function’s current size.

Basic Concepts of Logarithms

A logarithm is the inverse operation to exponentiation.
Just as exponentials are used to express very large numbers, logarithms are useful for expressing very small numbers.
The logarithm to the base ‘b’ of ‘a’ is denoted as logb(a), meaning the number we need to raise ‘b’ to get ‘a’.
The number ‘b’ is referred to as the base of the logarithm.
A special case is the natural logarithm, which is logarithm to the base ‘e’, and is usually written as ln(x) instead of loge(x).

Properties of Exponentials

The product rule states that a^x * a^y = a^(x+y).
The quotient rule postulates that a^x / a^y = a^(x-y).
The power of a power rule postulates that (a^x)^y = a^(xy).

Properties of Logarithms

The ‘log of a product’ becomes the sum of logs: logb(xy) = logb(x) + logb(y).
The ‘log of a quotient’ becomes the difference of logs: logb(x/y) = logb(x) - logb(y).
The ‘log of a power’ becomes the power times log: logb(x^n) = n * logb(x).
The log base b of b is 1, i.e. logb(b) = 1.

Remember that understanding these basic principles about exponentials and logarithms, and how to manipulate them, is key to tackling various types of problems in mathematics.