Exponential and logarithmic functions are powerful tools for representing and analyzing various phenomena in mathematics, science, and finance. These functions have distinct properties and applications which make them essential to understand.

Exponential Functions: Exponential functions can be written in the form f(x) = a^x, where 'a' is the base and 'x' is the exponent. They exhibit rapid growth or decay rates. Some key properties of exponential functions are:

• As the exponent increases, the function grows exponentially.
• If a > 1, the function is positively increasing. If 0 < a < 1, the function is exponentially decreasing.

An example of an exponential function is f(x) = 2^x. As x increases, the values of f(x) grow rapidly.

Logarithmic Functions: Logarithmic functions are the inverse of exponential functions. They can be written in the form f(x) = log_a(x), where 'a' is the base and 'x' is the argument. Logarithmic functions have unique properties:

• The logarithm of a product is the sum of the logarithms of the individual factors.
• The logarithm of a quotient is the difference of the logarithms of the numerator and denominator.

For instance, log_2(8) + log_2(4) = log_2(32).

Applications: Exponential and logarithmic functions find applications in various fields. They are used to model population growth, compound interest, radioactive decay, and more. For example, the exponential function can represent the growth of a bacteria population over time, while the logarithmic function can determine the time it takes for a material to decay to a certain level.

Embrace the power of exponential and logarithmic functions as you explore their properties and applications. Keep practicing and remember, you've got this!