Modelling with ex and In x

Understanding Exponential Modelling

Exponential models can be represented in the form y = ab^x where a is the initial amount and b is the growth factor.
The base, b is always positive and a number different than one. If b > 1, we have exponential growth, and if 0 < b < 1, we have exponential decay.
Exponential models can be manipulated to solve for unknown variables using logarithms.
An exponential graph crosses the y-axis at the point (0, a).

The number e (approximately 2.71828) is called Euler’s number and is an important mathematical constant.
The function y = e^x is the standard exponential function, and its graph passes through the point (0, 1).
This function increases rapidly for positive x and tends towards zero for negative x.
When modelling real-world problems, use y = ae^kx where a is the initial amount, k is the exponential growth (or decay) constant, and x is time.

ln x is the natural logarithm of x, and it is the inverse function of e^x.
Key properties of logarithms, such as ln(ab) = ln a + ln b and ln(a^b) = b ln a, are instrumental in simplifying expressions and solving equations.
The logarithm of a number less than 1 is negative, and the logarithm of 1 is 0.
Logarithms can be used to solve exponential equations by taking the natural log of both sides to isolate the variable.
Real-world models often use the form y = ln(x/a) / k where a is a scaling factor and k is a rate constant.

Remember to practise regularly to solidify these concepts and being able to use them in complex problem-solving. Practice makes perfect!