Finding inverses of exponential and logarithmic functions.

Index laws, log laws, equivalent forms, inverse functions. Assumed skills. Solving exponential equations. Log laws and solving log equations. Index laws, log laws, equivalent forms, inverse functions.

To find the inverse of a logarithmic function, follow these steps: Example: Find the inverse of Step 1: write as Step 2: swap x and y: Step 3: make y the subject by changing to an exponential form first : Step 5: replace y with Step 6: write down the final answer Step 4: then subtract 3 from both sides:

To find the inverse of an exponential function, follow these steps: Example: Find the inverse of Step 1: write as Step 5: add 1 to both sides Step 2: swap x and y: Step 3: make y the subject by subtracting 5 from both sides first : Step 5: replace y with Step 6: write down the final answer Step 4: then take log base 3 of both sides:

Draw both the original function and the inverse function on your calculator to check the answer graphically:

Example: Find the inverse of Check your solution by graphing the function and its inverse that you found and verifying visually that the reflection property holds. ANSWER:

Finding an inverse function using TI Nspire CAS In a calculator screen: define f(x), then use Algebra solve (x=f(y),y) as shown in the screen below:

Finding an inverse function using TI Nspire CAS In a graph screen: enter f(x) in Graph screen, graph entry function. Then change graph entry to relation and type x=f(y)

Practice time:

Visualisation of the correct solution in the last practice question.

Verify by composition that the given functions are inverses. Either Or:

Summary: As each exponential function and logarithmic function is one-to-one, they will always have an inverse function. To find the inverse functions graphically, reflect the original function in the line y=x. To find the equation for the inverse function, start by swapping x and y in the original function; then apply index laws and log laws to make y the subject. Finally, the inverse of exponential is logarithmic and vice versa. Remember to write your answer as