Final answer:
The logarithmic equation y = log_3 x is rewritten in exponential form as 3^y = x. Graphing this function with integer y-values from -2 to 2 shows an exponential growth pattern, becoming steeper as x increases.
Step-by-step explanation:
Rewriting a Logarithmic Equation in Exponential Form and Graphing it
To rewrite the logarithmic equation y = log_3 x in exponential form, we use the definition of a logarithm. The equation states that 3 raised to the power of y equals x. Therefore, the exponential form of the given logarithmic equation is 3^y = x.
To graph this function, we choose integer values for y within the range of -2 to 2 and solve for x to find corresponding values. For y=-2, we have 3^(-2) which gives x=1/9. Similarly, for y=-1, x=1/3; for y=0, x=1; for y=1, x=3; and for y=2, x=9.
Plotting these points on a coordinate system, we notice that the graph of 3^y (or log_3 x) increases rapidly as y increases, known as exponential growth. The plot passes through (1/9, -2), (1/3, -1), (1, 0), (3, 1), and (9, 2) and is a curve that gets steeper as we move to the right on the x-axis.
It's important to note the behavior of the function: as x approaches zero from the right, y approaches negative infinity, which represents the asymptotic nature of the exponential function. Meanwhile, for increasing positive x, the values of y increase without bound.