CP Math Lesson 10-3 Inverses of Logarithmic and Exponential functions.

Finding the Inverse of a function. y = x – 2 Exchange ‘x’ and ‘y’ Add ‘2’ (left and right) is the inverse of: y = x - 2 Solve for ‘y’

Extremely important idea. Find the inverse of f(x) Exchange ‘x’ and ‘y’ Power base 3  log base 3 are inverse functions “undo” power base 3  log base 3

Extremely important idea. Find the inverse of f(x) Exchange ‘x’ and ‘y’ Power base 3  log base 3 are inverse functions “undo” power base 2  log base 2

Exchange ‘x’ and ‘y’ How do you find an inverse function? Exchange ‘x’ and ‘y’ then solve for ‘y’ Divide left/right by 3 “undo” power base 4 Add ‘1’ left/right

Your Turn: Find for each of the following” 15.

Your Turn: Find 16.

Find 17.

Extremely important idea. Find the inverse of f(x) Exchange ‘x’ and ‘y’ Power base 3  log base 3 are inverse functions Isolate the log, undo the log “undo” log base 3  power base 3

Extremely important idea. Find the inverse of f(x) Exchange ‘x’ and ‘y’ Power base 3  log base 3 are inverse functions Isolate the log, undo the log Divide left/right by 2 “undo” log base 3  power base 3 Add ‘1’,

Your Turn: Find for each of the following”

Simplifying Logarithmic Expressions using inverse functions Your turn:

Solve equations using inverse functions.

“isolate the square, undo the square” ÷2 ÷2 Isolate the base and its exponent. exponent. “undo” squaring Use the inverse of the square function. Even Root !!!  ± result x = 6, 2 Do the math!!!

Key point: To solve an equation you must know how to “undo” a function. To solve this you must “undo” a square function. To solve this you must “undo” a square root function. To solve this you must “undo” a 5/2 power.

Your turn: Solve each of the following:

How do you solve the following function for ‘x’?

Logarithm Function Exponential Function reflections of each other across the line y = x. They are inverses of each other.

How do you solve the following function for ‘x’? These are inverse functions of each other. You must “undo” a power base 3. Log base 3 “un-does” power base 3 !!!!

Extremely important idea. Power base 3  log base 3 are inverse functions

Solve for ‘x’ Log base 2 is the inverse of power base 2 !!!!

Your turn: “undo” each of the following to solve for x:

Change-of-Base Formula for Logarithms Change of Base Formula: Change to log base 10 or base ‘e’ (your calculator can do these). (your calculator can do these). Convert to base 10.

Another Example: Calculate the following using the base conversion formula. base conversion formula.

Solve for ‘x’ Log base 2 is the inverse of power base 2 !!!! Check your solution

Your turn: Calculate ‘x’ using the base conversion formula. base conversion formula

Your turn. 9. Solve the following. 10.

Solve for ‘x’ “Isolate the power, “undo” the power” Divide by 2 Undo power base 4 with log base 4 Change of base formula Add 3

Your turn. 11. Solve the following. 12.

Your turn. 13. “isolate the power, “undo” the power”

Your turn. “isolate the power, “undo” the power” 14.

Real World Logarithmic Model Formula relating distance (d) that a tornado travels and the wind speed (s) inside the cone of the tornado. and the wind speed (s) inside the cone of the tornado. In 1925, a tornado traveled 220 miles through 3 states. Estimate the wind speed inside the tornado. (1) Write formula (2) Plug numbers into the formula (3) Solve for the unknown variable in the formula

Your turn: Formula relating distance (d) that a tornado travels and the wind speed (s) inside the cone of the tornado. and the wind speed (s) inside the cone of the tornado. 23. A tornado traveled 100 miles. Estimate the wind speed inside the tornado. (1) Write formula (2) Plug numbers into the formula (3) Solve for the unknown variable in the formula

Real World Logarithmic Model What if the problem was: “The wind speed is 200mph, how far will the tornado travel on the ground? “The wind speed is 200mph, how far will the tornado travel on the ground? (1) Write formula (2) Plug numbers into the formula (3) Solve for the unknown variable in the formula (remember: “isolate the log, undo the log”)

Real World Logarithmic Model “The wind speed is 200mph, how far will the tornado travel on the ground? Subtract 65 from both sides Divide by 93 (both sides) remember: “isolate the radical” ?  same thing, “isolate the log” We need the “inverse function” of log (base 10). log (base 10). Exponential base 10 We will solve log and exponential equations later.

Simplifying Logarithmic Expressions using inverse functions Can I rewrite 27 as a power of 3? Can I rewrite 8 as a power of 2?

Your turn:

Find the inverse function of: Convert to exponential form 29. What is the base? Convert to logarithm form

Inverse function of a logarithm Inverse of a log (base 4) is an exponential (base 4) Inverse of a log is an exponential function. Compose a function with its “inverse function” to “undo” the original function. original function. Log (base 4) of an exponential (base 4)

Inverse function of a logarithm Inverse of a log (base e) is an exponential (base e) Inverse of a log is an exponential function. Compose a function with its “inverse function” to “undo” the original function. original function. Log (base e) of an exponential (base e)

Inverse function of a logarithm Inverse of an exponential (base 4) is a log (base 4) Inverse of an exponential function is a logarithm function. Compose a function with its “inverse function” to “undo” the original function. original function. Exponential (base 4) raised to the log (base 4)

Inverse function of a logarithm Inverse of an exponential (base e) is a log (base e) Inverse of an exponential function is a log function. Compose a function with its “inverse function” to “undo” the original function. original function. exponential (base e) raised to the log (base e)

Finding the Inverse Function How do you find the inverse of: (1) Exchange ‘x’ and ‘y’ (2) Solve for ‘y’ How do you find the inverse of: (1) Exchange ‘x’ and ‘y’ (2) Solve for ‘y’ (compose with its inverse function) its inverse function) (compose with its inverse function) its inverse function)

Your Turn: What is the base? Convert to an exponential Convert to a logarithm