Functions and Logarithms

One-to-One Functions A function f(x) is one-to-one if f(a) ≠ f(b) whenever a ≠ b. Must pass horizontal line test. Not one-to-one One-to-one

Inverses If a function is one-to-one, then it has an inverse. ◦ Notation: f -1 (x) ◦ IMPORTANT:

Inverses To find the inverse of a function, solve for x, “swap” x and y, and solve for y. Ex: What is the inverse of f(x) = -2x + 4? y = -2x + 4 y – 4 = -2x (“swap” x and y) Therefore,

Inverses Inverse functions are symmetric (reflected) about the line y = x. Therefore, in order for two functions to be inverses, the results of the composites is x.

Inverses Example: Prove the two functions from our last example are inverses. = x Now, we must check the other composite! = x Therefore, these two functions must be inverses of one another!!!

Logarithms y = log a x means a y = x ◦ Ex: log 3 81 = 4 means 3 4 = 81 What is the inverse of y = log a x? Since y = log a x is a y = x, then the inverse has to be a x = y

Logarithms Common logarithms: log x means log 10 x ln x means log e x

Inverse Properties of Logs (both of these hold true if a > 1 and x > 0) (both of these hold true if x > 0)

Inverse Properties of Logs Example: Solve ln x = 3t + 5 for x. (use each side as an exponent of e) (e and ln are inverses and “undo” each other.)

Inverse Property of Logs Example: Solve e 2x = 10 for x. (take the natural log of both sides) (ln and e are inverses and “undo” each other.)

Properties of Logarithms For any real numbers x > 0 and y > 0,

Change of Base Property Since our calculators will not calculate logs of bases other than 10 or e,

Example Sarah invests $1000 in an account that earns 5.25% interest compounded annually. How long will it take the account to reach $2500? (divide both sides by 1000) (take a log of both sides…doesn’t matter what base you use!!!) (by my property, exponent comes out front) (divide by ln(1.0525)) t ≈ years