Understanding Exponential and Logarithmic Functions
TLDR; Matching graphs of exponential and logarithmic functions, including base 10 and e, involves identifying the growth rate and inverse relationships. Symmetry across the line y=x helps to identify inverse functions.
📈 Exponential Functions and Growth
The functions y = 10 to the power of x and y = e to the power of x represent exponential growth with bases greater than 1, indicating faster growth for y = 10 to the power of x due to its base being greater than that of y = e to the power of x.
The red graph corresponds to y = 10 to the power of x, while the blue graph matches y = e to the power of x, as the former increases faster from left to right.
By finding specific points for each function, it becomes evident that the exponential functions have been correctly identified.
📊 Matching Points to Functions
For y = 10 to the power of x, the point (1,10) is identified, while for y = e to the x, the point (1,e) is established.
Matching these points to the respective graphs confirms the correct identification of the exponential functions.
🔄 Inverse Functions
The functions y = 10 to the power of x and x = 10 to the power of y are recognized as inverse functions, as are y = e to the x and x = e to the y.
This is evident from the interchanged x and y variables in the equations.
Similarly, y = log x and x = 10 to the power of y are inverse functions, along with y = natural log x and x = e to the y.
The points (10,1) and (e,1) satisfy the inverse function equations, leading to the correct matching of the logarithmic functions with the graphs.
🔍 Graphing Inverse Functions
Graphing the line y = x helps identify the symmetry of inverse functions. The blue and black functions, representing y = e to the x and y = natural log x, are symmetrical across the line y = x, as are the green and red functions corresponding to y = 10 to the x and y = log x.
This symmetry across the line y = x assists in correctly matching the graphs with the logarithmic and exponential functions.