For very steep functions, it is possible to plot points more smoothly while retaining the integrity of the data: one can use a graph with a logarithmic scale, where instead of each space on a graph representing a constant increase, it represents an exponential increase.

Explain the relationship between exponential and logarithmic functions. Describe how to calculate a logarithm to a different base. Identify the hyperbolic functions, their graphs, and basic identities. In this section we examine exponential and logarithmic functions.

Exponential and logarithmic functions are inverse operations of each other. Exponential functions have a base raised to a variable exponent, while logarithmic functions have a base raised to a constant exponent equal to the variable.

A logarithmic function of the form [latex]y=log{_b}x[/latex] where [latex]b[/latex] is a positive real number, can be graphed by using a calculator to determine points on the graph or can be graphed without a calculator by using the fact that its inverse is an exponential function.

Graphs of Exponential \displaystyle {y}= {b}^ {x} y = bx, and Logarithmic \displaystyle {y}= { {\log}_ { {b}} {x}} y = logbx Functions. We saw an example of an exponential growth graph (showing how invested money grows over time) at the beginning of the chapter. The exponential curve is especially important in mathematics.

Exponential functions have a distinctive shape: as x x increases, the function grows or decays at an increasing rate, depending on whether a > 1 a> 1 or 0 < a < 1 0 <a <1. In other words, the function increases or decreases very rapidly as x x moves away from zero.

The inverse of an exponential function is a logarithmic function. The inverse of a function can be thought of as the function obtained by exchanging the $x$ and $y$ coordinates. This reflects the graph about the line $y=x$. As seen from the graph in Fig 2, the logarithmic curve (blue) is the reflection of the exponential curve (red).

Logarithmic Function – Any function in the form of y = logax which is the exponent y such that 𝑎 =𝑥. The number a is called the base of the logarithm and a can be any positive constant other than 1. Example: Graph the following logarithmic function by using a table to find at least three ordered pairs. 3 ; f :x ;= log2x

Explain the relationship between exponential and logarithmic functions. Describe how to calculate a logarithm to a different base. Identify the hyperbolic functions, their graphs, and basic identities. In this section we examine exponential and logarithmic functions.

Graphing Logarithmic Functions The function y = log b x is the inverse function of y = b x . So, it is the reflection of that graph across the diagonal line y = x . When no base is written, assume that the log is base 10 .