Logarithmic and Exponential Functions

• log_a y = x ⇔ a^x = y. The laws: log(xy) = log x + log y, log(x/y) = log x − log y, log x^k = k log x.
• ln x = log_e x. e^x and ln x are inverse functions: e^{ln x} = x and ln(e^x) = x.
• y = e^kx is growth for k > 0 and decay for k < 0; both pass through (0, 1).
• Take logs of both sides to solve equations or inequalities with the unknown in an index.
• y = kxⁿ becomes ln y = ln k + n ln x; y = k(a^x) becomes ln y = ln k + x ln a.

Indices written another way

Every law of logarithms is a law of indices in disguise. The laws hold for any base, including e. Change of base is not required for this paper.

The graph of y = e^x passes through (0, 1), is always positive and has the x-axis as an asymptote. y = ln x is its reflection in y = x: it passes through (1, 0), is defined only for x > 0 and has the y-axis as an asymptote. For y = e^kx: k > 0 rises steeply; k < 0 falls towards the x-axis.

Unknown indices and straight-line graphs

To solve a^f(x) = b^g(x), take ln of both sides: f(x) ln a = g(x) ln b, a linear equation when f and g are linear. For an inequality, the same step is safe because ln is an increasing function, but dividing by a negative log (such as ln 0.5) reverses the sign.

To find constants in a model, take ln of both sides to get the form Y = mX + c, then read the gradient and intercept from the straight-line graph.