Exponentials and Logs

Basics of Exponentials and Logarithms

Exponential functions are functions expressed as y = a * b^x, where ‘a’ and ‘b’ are constants, ‘b’ differs from 1 and x is a variable.
Logarithmic functions are the reverse of exponential functions and can be written as y = log_b (x), where ‘b’ is the base of the logarithm and ‘x’ is a positive number.
The number ‘e’ is an important mathematical constant which is approximately equal to 2.71828.
The function y = e^x is often used as the base exponential function in calculus.

Laws of logarithms include the product rule: log_b(a*c) = log_b(a) + log_b(c).
The quotient rule: log_b(a/c) = log_b(a) - log_b(c).
The power rule: log_b(a^n) = n * log_b(a).
Laws of exponentials: (a^m)^n = a^(mn), a^m * a^n = a^(m+n), a^m / a^n = a^(m-n), a^0 = 1.

To change the base of a logarithm, use the Change of Base formula: log_a(b) = log_c(b) / log_c(a), where c can be any positive number - usual choices being 10, e or another convenient number.

The graph of y = a * b^x is always above the x-axis (y > 0) because any number raised to a power is always positive.
The graph of y = log_b(x) has its domain as all positive numbers and range as all real numbers. The function has a vertical asymptote at x = 0.
The graphs of y = e^x, y = ln(x), y = log10(x) and y = 10^x are particularly important.

To solve exponential equations, you may need to use logarithms to find the value of x.
Logarithmic equations can often by solved by rewriting the equation in exponential form.
Always check solutions to logarithmic equations as the domain of the original logarithmic function may restrict the possible values of the variables.
Be careful when simplifying expressions involving exponentials and logarithms, remember to apply the laws accurately and in the appropriate order.

Exponential functions can be used to model real-life situations such as population growth or radioactive decay.
The general form of an exponential growth or decay function is y = a * (1 ± r)^t where a is the initial amount, r is the rate of growth/decay, t is time, and the ‘+’ sign is used for growth and ‘–’ for decay.
You can also use the continuous growth model y = a * e^(rt), especially when dealing with continuously compounding interest rates.

The natural logarithm, or ln, is a special case where the base of the logarithm is e. It plays a crucial role especially in calculus.
Understanding the relationship between e, ln, and their graphs is necessary for differentiation and integration involving exponentials and logarithms.

Study these points, practise using them in examples, and the formal properties and rules which go with them will soon become second nature.