What is the difference between exponential and logarithmic functions?

What is the difference between exponential and logarithmic functions?

Logarithmic functions are the inverses of exponential functions. The inverse of the exponential function y = ax is x = ay. The logarithmic function y = logax is defined to be equivalent to the exponential equation x = ay. So you see a logarithm is nothing more than an exponent.

What is the difference between logarithmic and exponential growth?

The logarithm is the mathematical inverse of the exponential, so while exponential growth starts slowly and then speeds up faster and faster, logarithm growth starts fast and then gets slower and slower.

How do you determine if a function is logarithmic or exponential?

The inverse of an exponential function is a logarithmic function. Remember that the inverse of a function is obtained by switching the x and y coordinates. This reflects the graph about the line y=x. As you can tell from the graph to the right, the logarithmic curve is a reflection of the exponential curve.

How are exponential and logarithmic functions used in real life?

Exponential and logarithmic functions are no exception! Much of the power of logarithms is their usefulness in solving exponential equations. Some examples of this include sound (decibel measures), earthquakes (Richter scale), the brightness of stars, and chemistry (pH balance, a measure of acidity and alkalinity).

What are the characteristics of logarithmic functions?

Characteristics of Logarithmic Functions Graphs

  • The graph of logarithmic functions passes through the points (1,0).
  • If the base of a logarithmic function is greater than 1, then the graph increases.
  • If the base of the logarithmic functions is greater than 0 but smaller than 1, then the graph decreases.

Is logarithmic the opposite of exponential?

The logarithmic function g(x) = logb(x) is the inverse of the exponential function f(x) = bx.

Why is a logarithmic function the inverse of an exponential function?

Since g(x) = logb x is the inverse function of f(x) the domain of the log function will be the range of the exponential function, and vice versa. So the domain of logb x is (0,∞) and the range is (−∞,∞).

What are the characteristics of exponential functions?

Graphs of Exponential Functions

  • The graph passes through the point (0,1)
  • The domain is all real numbers.
  • The range is y>0.
  • The graph is increasing.
  • The graph is asymptotic to the x-axis as x approaches negative infinity.
  • The graph increases without bound as x approaches positive infinity.
  • The graph is continuous.

How are exponential functions graphed?

A simple exponential function to graph is y=2x . Replacing x with −x reflects the graph across the y -axis; replacing y with −y reflects it across the x -axis. Replacing x with x+h translates the graph h units to the left.

What is the difference between exponential and logarithmic differentiation?

Differentiation of Exponential and Logarithmic Functions Exponential functions and their corresponding inverse functions, called logarithmic functions, have the following differentiation formulas: Note that the exponential function f (x) = e x has the special property that its derivative is the function itself, f ′ (x) = e x = f (x).

What are the exponential and logarithm functions in calculus?

The most common exponential and logarithm functions in a calculus course are the natural exponential function, ex e x, and the natural logarithm function, ln(x) ln

How to find the derivative of a general logarithm function?

In this case, unlike the exponential function case, we can actually find the derivative of the general logarithm function. All that we need is the derivative of the natural logarithm, which we just found, and the change of base formula. Using the change of base formula we can write a general logarithm as, Differentiation is then fairly simple.

How to find the value of f (x) in exponential form?

For the natural exponential function, f (x) = ex f (x) = e x we have f ′(0) = lim h→0 eh−1 h = 1 f ′ (0) = lim h → 0 e h − 1 h = 1. So, provided we are using the natural exponential function we get the following. f (x) =ex ⇒ f ′(x) = ex f (x) = e x ⇒ f ′ (x) = e x