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The Graph of a Logarithmic Function

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- Graph of Exponential Function versus Graph of Logarithmic Function
- Domain and Range of Logarithmic Functions
- x-Intercept of Logarithmic Functions

**Graph of Exponential Functions versus Graph of Logarithmic Functions**

Exponents and logarithms are inverse operations. As functions, they are also inverses of each other. There is a special relationship between inverse functions on a graph: the line y = x is a line of symmetry or reflection between a function and its inverse. So we should expect this relationship to hold true when examining the graphs of an exponential function and a logarithmic function:

**Domain and Range of Logarithmic Functions**

As we can see in the graph above, since a function is symmetrical about the line y = x to its inverse, we can swap x– and y–values to plot points on a function's inverse. This also means that the domain and range of a function switches as we describe the domain and range of its inverse. That is, the domain of a function becomes the range of its inverse, and the range of a function becomes the domain of its inverse.

Looking at the graph of the exponential function, we can see that the domain is all x-values. This means that the range of logarithmic functions is all y values, from negative infinity to positive infinity. However, when looking at the range of the general exponential function graphed above, we see that the range is from zero to positive infinity. There are no x values that make y negative (at least when both "a" and "b" are positive). This means that the domain of the logarithmic function is restricted to zero through positive infinity. Inputting a negative value into the logarithmic function will yield a non-real answer.

The domain and range of a logarithmic function is the range and domain of an exponential function:

- The domain of log functions is restricted to x values of zero or greater.
- The range of log functions is all real numbers

**x-Intercept of Logarithmic Functions**

When studying exponential functions, you may recall that the y-intercept of y = b^{x} is at (0, 1) on the graph, because when x = 0, y = 1. This is because any value raised to the power of zero evaluates to 1. When comparing this to logarithmic functions, we invert x and y (maybe you are starting to see a pattern here!). So instead of talking about the y-intercept of an exponential function, we talk about the x-intercept of a logarithmic function; and instead of having coordinates of (0, 1), it has the coordinates (1, 0). This is illustrated in the graph below: