To find the inverse of a function (the inverse of a function is the same as reflecting a function across the line y = x), interchange x and y and then solve for y. The inverse of f(x) is denoted by f-1(x). Example:
1. Problem: Find f-1(x) of 3x + 1 Solution: The equation is y = 3x + 1. Interchange x and y. x = 3y + 1 Solve for y. x - 1 = 3y (x - 1)/3 = y f-1(x) = (x - 1)/3
Exponential functions are functions where f(x) = ax + B where a is any real constant and B is any expression. For example, f(x) = e-x - 1 is an exponential function.
1. Problem: Graph f(x) = 2x. Solution: Plug in numbers for x and find values for y, as we have done with the table below. _____________________ | x | 0 | 1 | 2 | 3 | --------------------- | y | 1 | 2 | 4 | 8 | --------------------- Now plot the points and draw the graph (shown below).
Logarithmic functions are the inverse of exponential functions. For example, the inverse of y = ax is y = logax, which is the same as x = ay.
1. Problem: Convert to logarithmic form: 8 = 2x Solution: Remember that the logarithm is the exponent. x = log2 8 2. Problem: Convert to exponential form: y = log3 5 Solution: Remember that the logarithm is the exponent. 3y = 5The figure below is a little chart that always helped us remember how to convert from exponential to logarithmic form and from logarithmic to exponential form.
3. Problem: Solve log2 x = -3. Solution: Convert the logarithm to exponential form. 2-3 = x x = (1/8)
There are five special rules that you ought to always have in mind when working with logarithms. They will help you in such tasks as simplifying expressions containing logarithms and solving equations containing logarithms. They are outlined below.
4. Problem: Simplify: log2 x + log2 6. Solution: log2 (x * 6)
2. For any positive numbers x and p, loga xp = p * loga x. Example:
5. Simplify: logb 9-x Solution: -x * logb 9
3. For any positive numbers x and y, loga (x/y) = loga x - loga y. Example:
6. Problem: Express as a single logarithm: loga x - 5loga y Solution: loga x - loga y5 (Using the 2nd rule.) Use the third rule in reverse. loga (x/y5)
4. loga a = 1
An equation with variables in its exponents is called an exponential equation. To solve these, take logarithms of both sides and use theorems 1 - 5 listed in the section above to simplify and then solve for x. Example:
1. Problem: Solve for x: 3x = 8. Solution: Take the logarithm of both sides. log 3x = log 8 Use theorem 2 to simplify the equation. x * log 3 = log 8 Solve for x by dividing each side by log 3. x = (log 8/log 3) A decimal approximation may be found if desired - x = 1.8929.
To solve logarithmic equations, you convert them to exponential form and solve for x. Example:
2. Problem: Solve log3 (5x + 7) = 2 for x. Solution: Write an equivalent exponential expression. 5x + 7 = 32 5x + 7 = 9 Solve for x. 5x = 2 x = (2/5)