As you've seen, transcendental functions are fundamental in mathematics and science, describing exponential growth, logarithmic scaling, and periodic behavior. In this section, you'll skilfully use Python to help visualize these functions dynamically.

Exponential Function

Exponential functions model rapid growth or decay, commonly used in population modeling, finance, and physics. This function is of the form $f(x) = ae^{bx}$.

Code Breakdown

• Generates x values between -5 and 5;
• Defines exponential_function(x, a, b), where a scales the function, and b controls the growth rate;
• Plots the graph with arrows at both ends to show continuous growth;
• Marks the y-intercept at x = 0 for clarity.

Logarithmic Function

Logarithms are the inverse of exponentials, useful in scaling data and measuring natural growth processes. This function is defined as $f(x) = \log_2(x)$, meaning it calculates the power to which $2$ must be raised to obtain $x$.

Code Breakdown

• Generates x values between 0.1 and 10 (to avoid log(0), which is undefined);
• Defines logarithmic_function(x, base=2), ensuring base 2 is used throughout;
• The graph includes an arrow at the right end, indicating it continues indefinitely;
• The x-intercept is marked at x = 1, where log_2(1) = 0.