Quiz 7-5: Expand Condense Use these to find: Use change of base to solve 7.

7-6 Solve Exponential and Logarithmic Equations

What you’ll learn Review of Inverse Functions … and why Logarithms are used extensively in science. Solving equations is how problems in science are solved. Using Logarithms to solve exponential equations Property of Equality Using exponents to solve logarithmic equations Real World Problems: Newton’s Law of cooling Distance a tornado stays on the ground The apparent “magnitude” of a star

Your turn: 1. Use the change of base formula to find: 2.

Solving Equations Review Radical Equation: “Isolate the radical” “Undo the radical” add 1 to both sides +1+1 square both sides add 2 to both sides +2+2 divide both sides by 3 sides by 3

Solving an Exponential Equation: The easiest problem “Isolate the power, Undo the power” What is the inverse function of power base 2? of power base 2? Logarithm exponent property Remember:

Another example “Isolate the power, Undo the power” Apply inverse function of “power base 7” base 7” Logarithm exponent property Remember: +4x +4x -1 -1

Your turn: Solve : 4. “Isolate the power” then “undo the power” “undo the power” 3.

Changing the base of a power Easy Harder

Your turn: Change the base of the power as indicated :

Solving using “convert to same base” “Undo the exponent” If I take log base 9 of each side it won’t eliminate BOTH bases Can the two bases be rewritten as a power of the same base? a power of the same base? Power of a power rule Distributive Property Property of equality

Solving using “power” of the power rule Take natural log of both side power rule simplify ÷ ln 9 ÷ ln 9 simplify -1.5x -1.5x x by 2 x by 2

8 and 4 are both powers of 2. Power of a Power property “undo the power” -2x -2x Did I do a step mentally? Solving using “convert to same base”

Take natural log of both sides Power property x x Solving using the “power” of the power rule ÷ ln 8 ln 8 ÷ ÷0.333

Your turn: Solve : 9. “Isolate the power” then “undo the power” “undo the power”

Solve using “undo the power” “Isolate the power” “Undo the power” Change of base formula formula ÷2 ÷2

Solve using the “power” of the power rule “Isolate the power” Natural log left/right ÷2 ÷2

Solve using “undo the power” “Isolate the power” “Undo the power” +2+2 Change of base formula formula

Solve using the “power” of the power rule “Isolate the power” Natural log left/right +2+2 Power rule

Your turn: Solve:

Natural Logarithm Function Natural Logarithm Function What is the domain? Does NOT make sense.

When solving logarithm equations Some solutions won’t make sense. Extraneous solution: an apparent solution that does not work when plugged back into the original equation. You MUST check the solutions in the original equation.

Solving Logarithmic Equations “Isolate the logarithm” “undo the logarithm” Inverse of log base 2 is exponent base 2. is exponent base 2. x = 7 Plug back in to check! Why do we need to check? Checks! Remember this: For a log equation: if the solution results in the log of a negative number, that number is NOT a solution. a negative number, that number is NOT a solution.

Solving Logarithmic Equations “Isolate the logarithm” “undo the logarithm” Inverse of log base 5 is exponent base 5. is exponent base 5. 4x - 7 = x + 5 3x = 12 x = 4 Subtract ‘x’ from both sides. Divide both sides by 3 Plug back in to check! Checks

Your Turn: Solve: Remember to check you solutions by plugging the solution for ‘x’ back into the original equation. for ‘x’ back into the original equation.

Solving Logarithmic Equations Power property of logarithms Change of base Use inverse property of multiplication

Your turn: Solve: “isolate the log” then “isolate the log” then “undo the log” “undo the log” Don’t forget the power property for logarithms. property for logarithms.

More complicated Logarithmic Equations “Isolate the logarithm” “undo the logarithm” Power property of logarithms Add ‘2’ to both sides. Change of base Use inverse property of multiplication

Your turn: Solve: “isolate the log” then “isolate the log” then “undo the log” “undo the log” Don’t forget the power property for logarithms. property for logarithms.

Solving Logarithmic Equations “Isolate the logarithm” “undo the logarithm” Inverse of log base 4 is exponent base 4. is exponent base 4. 5x - 1 = 64 Subtract ‘1’ from both sides Divide both sides by ‘5’ x = 13 Plug back in to check! Checks

Your Turn: 19. Solve: 20.

Solving Logs requiring condensing the product. “Isolate the logarithm” “undo the logarithm” “condense the product” 2x(x - 5) = 100 Inverse of log base 10 is exponent base 10 is exponent base 10 Quadratic  put in standard form Divide both sides by ‘2’ factor x = 10, -5 Zero factor property

Check the solution: x = 10, -5 “Condense the product” Convert to exponent form Checks

Check the solution: x = 10, -5 Or, convert to exponent form This doesn’t make sense  punch this into your calculator. this into your calculator. There is NO exponent that will cause a positive number to equal a negative number. positive number to equal a negative number. -5 is NOT a solution a solution

Your Turn: “Undo the exponent”

Using this idea to solve equations Replace 36 with a power with base ‘6’ : Power of a Power property “undo the power” -x -x -2 -2

Newton’s Law of Cooling A high temperature item will cool off in a lower temperature medium in which it is placed. This cooling off process can be modeled by the following equation. Temperature (as a function of time) Surrounding Temperature Temperature Initial Temp of the object of the object Cooling rate Time

Your Turn: You are baking a cake. When you take the cake out of the oven, it is at 350ºF. The room temperature is 70ºF. The cooling rate is How many minutes will it take for the cake cool to 100ºF? 23. Solve for ‘time’ Plug #’s into the formula. Isolate the power Undo the power Solve for ‘t’ t = 27.9 minutes t = 27.9 minutes

Example Newton’s Law of Cooling A hard-boiled egg at temperature 100 º C is placed in 15 º C water to cool. Five minutes later the temperature of the egg is 55 º C. r = 0.15 When will the egg be 25 º C? “isolate the power” “undo the base” Subtract 15 from both sides from both sides Divide both sides by 85 sides by 85 “undo log”  Natural log both sides Divide both sides by -0.15