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An exponential function takes the basic form of \( f(x) = a^x \), where \( a \) is a constant and \( x \) is the exponent. The graph of an exponential function like \( f(x) = 5^x \) can reveal a lot about these types of equations. Crucially, since \( a > 0 \), an exponential function will always yield positive outcomes, which means the graph will never touch the x-axis; the x-axis acts as an asymptote. The function swiftly increases (if \( a > 1 \)) as \( x \) grows.

To comprehend the graph, consider the point \( (0,1) \) a starting point. For each increase in \( x \), the value of \( f(x) \) multiplies by 5. For negative values of \( x \), the function approaches the asymptote at the x-axis, but without intersecting it. Visualization of exponential functions can be aided by plotting key points and noting the rapid growth as it moves to the right, and the fast approach towards zero to the left.

• The function is always above the x-axis due to the positive base.
• It has a horizontal asymptote at \( y = 0 \).
• The growth rate increases exponentially with increasing \( x \).

A logarithmic function is essentially the inverse of an exponential function. It's represented as \( g(x) = \log_b x \), where \( b \) is the base and the result is the power to which the base must be raised to obtain \( x \). The function \( g(x) = \log_5 x \), specifically, increases slowly and passes through the point \( (1,0) \).

Since the logarithmic function is the inverse of the exponential function, the two have related graphs. Notably, \( g(x) \) is undefined for \( x \leq 0 \), as logarithms are only defined for positive numbers. As \( x \) increases, \( g(x) \) does as well, but at a progressively slower pace. Here are some features that typify the logarithmic function graph:

• It has a vertical asymptote on the y-axis; the graph never touches or crosses the y-axis.
• It passes through \( (1,0) \) since any number to the power of zero equals one.
• It reflects the exponential function graph across the line \( y=x \).

The rectangular coordinate system, also known as the Cartesian coordinate system, is a 2D plane defined by a horizontal x-axis and a vertical y-axis intersecting at a right angle at the origin, which is denoted by \( (0,0) \). This system allows us to represent mathematical functions as visual graphs.

When graphing the exponential function \( f(x)=5^x \) and the logarithmic function \( g(x)=\log_5 x \), plotting points that satisfy each equation and drawing a smooth curve through the points help us visualize the behavior of the functions. Each point on the graph corresponds to a pair of \( (x, y) \) values that make the equation true. This system is fundamental not only for graphing these functions but for understanding their relationship to one another and their behavior within a visual context.

Function transformations involve altering a function's graph in various ways, such as shifting, stretching, compressing, and reflecting. By understanding these transformations, we can easily predict how the function's graph will change without plotting every single point.

For exponential and logarithmic functions, key transformations include vertical and horizontal shifts (translating the graph up, down, left or right), reflections across the axes (flipping the graph), and vertical stretching or compressing (changing the steepness of the graph).

• Reflecting \( f(x) = 5^x \) over the line \( y=x \) results in its inverse function \( g(x) = \log_5 x \).
• A horizontal translation of \( f(x) \) by \( c \) units is \( f(x-c) \), which shifts the graph to the right if \( c > 0 \) and to the left if \( c < 0 \).
• Vertical stretching is achieved by multiplying \( f(x) \) by a factor greater than 1, making the graph steeper.