Day 26

1. Opener.

Solve for x:

1. 10x = 5.712. 7e3x = 312

2. Exponential and Logarithmic Equations.

Exponential equations: An exponential equation is an equation containing a variable in an exponent.

Logarithmic equations: Logarithmic equations contain logarithmic expressions and constants.

Property of Logarithms, part 2:If x, y and a are positive numbers, a 1, then

If x = y, then

≠loga x = loga y

2. Exponential and Logarithmic Equations.

Example: Solve 2x+2 = 32x+1

2. Exponential and Logarithmic Equations.

Example: Solve 2x+2 = 32x+1

ln2x+2 = ln 32x+1

2. Exponential and Logarithmic Equations.

Example: Solve 2x+2 = 32x+1

ln2x+2 = ln 32x+1 Take the log of both sides

2. Exponential and Logarithmic Equations.

Example: Solve 2x+2 = 32x+1

ln2x+2 = ln 32x+1

x + 2( )ln2 = 2x +1( )ln 3Take the log of both sides

2. Exponential and Logarithmic Equations.

Example: Solve 2x+2 = 32x+1

ln2x+2 = ln 32x+1

x + 2( )ln2 = 2x +1( )ln 3Take the log of both sides

Property of Logarithms

2. Exponential and Logarithmic Equations.

Example: Solve 2x+2 = 32x+1

ln2x+2 = ln 32x+1

x + 2( )ln2 = 2x +1( )ln 3x ln2 + 2 ln2 = 2x ln 3+ ln 3

Take the log of both sides

Property of Logarithms

2. Exponential and Logarithmic Equations.

Example: Solve 2x+2 = 32x+1

ln2x+2 = ln 32x+1

x + 2( )ln2 = 2x +1( )ln 3x ln2 + 2 ln2 = 2x ln 3+ ln 3

Take the log of both sides

Property of Logarithms

Distributive property

2. Exponential and Logarithmic Equations.

Example: Solve 2x+2 = 32x+1

ln2x+2 = ln 32x+1

x + 2( )ln2 = 2x +1( )ln 3x ln2 + 2 ln2 = 2x ln 3+ ln 3x ln2 − 2x ln 3= ln 3− 2 ln2

Take the log of both sides

Property of Logarithms

Distributive property

2. Exponential and Logarithmic Equations.

Example: Solve 2x+2 = 32x+1

ln2x+2 = ln 32x+1

x + 2( )ln2 = 2x +1( )ln 3x ln2 + 2 ln2 = 2x ln 3+ ln 3x ln2 − 2x ln 3= ln 3− 2 ln2

Take the log of both sides

Property of Logarithms

Distributive property

Isolate terms (variable on one side of the equation).

2. Exponential and Logarithmic Equations.

Example: Solve 2x+2 = 32x+1

ln2x+2 = ln 32x+1

x + 2( )ln2 = 2x +1( )ln 3x ln2 + 2 ln2 = 2x ln 3+ ln 3x ln2 − 2x ln 3= ln 3− 2 ln2x ln2 − 2 ln3( ) = ln 3− 2 ln2

Take the log of both sides

Property of Logarithms

Distributive property

Isolate terms (variable on one side of the equation).

2. Exponential and Logarithmic Equations.

Example: Solve 2x+2 = 32x+1

ln2x+2 = ln 32x+1

x + 2( )ln2 = 2x +1( )ln 3x ln2 + 2 ln2 = 2x ln 3+ ln 3x ln2 − 2x ln 3= ln 3− 2 ln2x ln2 − 2 ln3( ) = ln 3− 2 ln2

Take the log of both sides

Property of Logarithms

Distributive property

Isolate terms (variable on one side of the equation).

Common factor, x.

2. Exponential and Logarithmic Equations.

Example: Solve 2x+2 = 32x+1

ln2x+2 = ln 32x+1

x + 2( )ln2 = 2x +1( )ln 3x ln2 + 2 ln2 = 2x ln 3+ ln 3x ln2 − 2x ln 3= ln 3− 2 ln2x ln2 − 2 ln3( ) = ln 3− 2 ln2

x = ln 3− 2 ln2ln2 − 2 ln3

Take the log of both sides

Property of Logarithms

Distributive property

Isolate terms (variable on one side of the equation).

Common factor, x.

2. Exponential and Logarithmic Equations.

Example: Solve 2x+2 = 32x+1

ln2x+2 = ln 32x+1

x + 2( )ln2 = 2x +1( )ln 3x ln2 + 2 ln2 = 2x ln 3+ ln 3x ln2 − 2x ln 3= ln 3− 2 ln2x ln2 − 2 ln3( ) = ln 3− 2 ln2

x = ln 3− 2 ln2ln2 − 2 ln3

Take the log of both sides

Property of Logarithms

Distributive property

Isolate terms (variable on one side of the equation).

Common factor, x.

Divide both sides by ln2 - 2ln3

2. Exponential and Logarithmic Equations.

Example: Solve log4 x + 3( ) = 2

2. Exponential and Logarithmic Equations.

Example: Solve log4 x + 3( ) = 2

42 = x + 3

2. Exponential and Logarithmic Equations.

Example: Solve log4 x + 3( ) = 2

42 = x + 3 Definition of Logarithm

2. Exponential and Logarithmic Equations.

Example: Solve log4 x + 3( ) = 2

42 = x + 3

16 = x + 3

Definition of Logarithm

2. Exponential and Logarithmic Equations.

Example: Solve log4 x + 3( ) = 2

42 = x + 3

16 = x + 3

Definition of Logarithm

Simplify

2. Exponential and Logarithmic Equations.

Example: Solve log4 x + 3( ) = 2

42 = x + 3

16 = x + 3

13= x

Definition of Logarithm

Simplify

2. Exponential and Logarithmic Equations.

Example: Solve log4 x + 3( ) = 2

42 = x + 3

16 = x + 3

13= x

Definition of Logarithm

Simplify

Solve for x.

2. Exponential and Logarithmic Equations.

Example: Solve log4 x + 3( ) = 2

42 = x + 3

16 = x + 3

13= x

Definition of Logarithm

Simplify

Solve for x.

All solutions of Logarithmic equations must be checked, because negative numbers do not have logarithms.

2. Exponential and Logarithmic Equations.

Example: Solve log2 x + log2 x − 7( ) = 3

2. Exponential and Logarithmic Equations.

Example: Solve log2 x + log2 x − 7( ) = 3log2 x x − 7( ) = 3

2. Exponential and Logarithmic Equations.

Example: Solve log2 x + log2 x − 7( ) = 3log2 x x − 7( ) = 3 Property of Logarithms

2. Exponential and Logarithmic Equations.

Example: Solve log2 x + log2 x − 7( ) = 3log2 x x − 7( ) = 323 = x x − 7( )

Property of Logarithms

2. Exponential and Logarithmic Equations.

Example: Solve log2 x + log2 x − 7( ) = 3log2 x x − 7( ) = 323 = x x − 7( )

Property of Logarithms

Definition of Logarithm

2. Exponential and Logarithmic Equations.

Example: Solve log2 x + log2 x − 7( ) = 3log2 x x − 7( ) = 323 = x x − 7( )8 = x2 − 7x

Property of Logarithms

Definition of Logarithm

2. Exponential and Logarithmic Equations.

Example: Solve log2 x + log2 x − 7( ) = 3log2 x x − 7( ) = 323 = x x − 7( )8 = x2 − 7x

Property of Logarithms

Definition of Logarithm

Simplify

2. Exponential and Logarithmic Equations.

Example: Solve log2 x + log2 x − 7( ) = 3log2 x x − 7( ) = 323 = x x − 7( )8 = x2 − 7x0 = x2 − 7x − 8

Property of Logarithms

Definition of Logarithm

Simplify

2. Exponential and Logarithmic Equations.

Example: Solve log2 x + log2 x − 7( ) = 3log2 x x − 7( ) = 323 = x x − 7( )8 = x2 − 7x0 = x2 − 7x − 8

Property of Logarithms

Definition of Logarithm

Simplify

Write cuadratic equation in standard form

2. Exponential and Logarithmic Equations.

Example: Solve log2 x + log2 x − 7( ) = 3log2 x x − 7( ) = 323 = x x − 7( )8 = x2 − 7x0 = x2 − 7x − 8

0 = x − 8( ) x +1( )

Property of Logarithms

Definition of Logarithm

Simplify

Write cuadratic equation in standard form

2. Exponential and Logarithmic Equations.

Example: Solve log2 x + log2 x − 7( ) = 3log2 x x − 7( ) = 323 = x x − 7( )8 = x2 − 7x0 = x2 − 7x − 8

0 = x − 8( ) x +1( )

Property of Logarithms

Definition of Logarithm

Simplify

Write cuadratic equation in standard form

Solve by factoring

2. Exponential and Logarithmic Equations.

Example: Solve log2 x + log2 x − 7( ) = 3log2 x x − 7( ) = 323 = x x − 7( )8 = x2 − 7x0 = x2 − 7x − 8

0 = x − 8( ) x +1( )

Property of Logarithms

Definition of Logarithm

Simplify

Write cuadratic equation in standard form

Solve by factoring

x = 8 or x = -1

2. Exponential and Logarithmic Equations.

Example: Solve log2 x + log2 x − 7( ) = 3log2 x x − 7( ) = 323 = x x − 7( )8 = x2 − 7x0 = x2 − 7x − 8

0 = x − 8( ) x +1( )

Property of Logarithms

Definition of Logarithm

Simplify

Write cuadratic equation in standard form

Solve by factoring

x = 8 or x = -1 Check!

2. Exponential and Logarithmic Equations.

Example: Solve log 2x −1( ) = log 4x − 3( )− log x

2. Exponential and Logarithmic Equations.

Example: Solve log 2x −1( ) = log 4x − 3( )− log x

log 2x −1( ) = log 4x − 3x

2. Exponential and Logarithmic Equations.

Example: Solve log 2x −1( ) = log 4x − 3( )− log x

log 2x −1( ) = log 4x − 3x

Property of Logarithms

2. Exponential and Logarithmic Equations.

Example: Solve log 2x −1( ) = log 4x − 3( )− log x

log 2x −1( ) = log 4x − 3x

2x −1= 4x − 3x

Property of Logarithms

2. Exponential and Logarithmic Equations.

Example: Solve log 2x −1( ) = log 4x − 3( )− log x

log 2x −1( ) = log 4x − 3x

2x −1= 4x − 3x

Property of Logarithms

Property of Logarithms

2. Exponential and Logarithmic Equations.

Example: Solve log 2x −1( ) = log 4x − 3( )− log x

log 2x −1( ) = log 4x − 3x

2x −1= 4x − 3x

Property of Logarithms

Multiply both sides by x

Property of Logarithms

2. Exponential and Logarithmic Equations.

Example: Solve log 2x −1( ) = log 4x − 3( )− log x

log 2x −1( ) = log 4x − 3x

2x −1= 4x − 3x

Property of Logarithms

Multiply both sides by x

Property of Logarithms

x(2x - 1) = 4x - 3

2. Exponential and Logarithmic Equations.

Example: Solve log 2x −1( ) = log 4x − 3( )− log x

log 2x −1( ) = log 4x − 3x

2x −1= 4x − 3x

Property of Logarithms

Multiply both sides by x

Distributive property

Property of Logarithms

x(2x - 1) = 4x - 3

2. Exponential and Logarithmic Equations.

Example: Solve log 2x −1( ) = log 4x − 3( )− log x

log 2x −1( ) = log 4x − 3x

2x −1= 4x − 3x

Property of Logarithms

Multiply both sides by x

Distributive property

Property of Logarithms

x(2x - 1) = 4x - 3

2x2 - x = 4x - 3

2. Exponential and Logarithmic Equations.

Example: Solve log 2x −1( ) = log 4x − 3( )− log x

log 2x −1( ) = log 4x − 3x

2x −1= 4x − 3x

Property of Logarithms

Multiply both sides by x

Distributive property

Property of Logarithms

x(2x - 1) = 4x - 3

2x2 - x = 4x - 3

2x2 - 5x + 3 = 0

2. Exponential and Logarithmic Equations.

Example: Solve log 2x −1( ) = log 4x − 3( )− log x

log 2x −1( ) = log 4x − 3x

2x −1= 4x − 3x

Property of Logarithms

Multiply both sides by x

Distributive property

Property of Logarithms

x(2x - 1) = 4x - 3

2x2 - x = 4x - 3

2x2 - 5x + 3 = 0 Write quadratic equation in standard form

2. Exponential and Logarithmic Equations.

Example: Solve log 2x −1( ) = log 4x − 3( )− log x

log 2x −1( ) = log 4x − 3x

2x −1= 4x − 3x

Property of Logarithms

Multiply both sides by x

Distributive property

(2x - 3)(x - 1) = 0

Property of Logarithms

x(2x - 1) = 4x - 3

2x2 - x = 4x - 3

2x2 - 5x + 3 = 0 Write quadratic equation in standard form

2. Exponential and Logarithmic Equations.

Example: Solve log 2x −1( ) = log 4x − 3( )− log x

log 2x −1( ) = log 4x − 3x

2x −1= 4x − 3x

Property of Logarithms

Multiply both sides by x

Distributive property

(2x - 3)(x - 1) = 0

Property of Logarithms

x(2x - 1) = 4x - 3

2x2 - x = 4x - 3

2x2 - 5x + 3 = 0 Write quadratic equation in standard form

Solve by factoring

2. Exponential and Logarithmic Equations.

Example: Solve log 2x −1( ) = log 4x − 3( )− log x

log 2x −1( ) = log 4x − 3x

2x −1= 4x − 3x

Property of Logarithms

Multiply both sides by x

Distributive property

(2x - 3)(x - 1) = 0

Property of Logarithms

x(2x - 1) = 4x - 3

2x2 - x = 4x - 3

2x2 - 5x + 3 = 0 Write quadratic equation in standard form

Solve by factoring

x = 3/2 and x = 1

2. Exponential and Logarithmic Equations.

Example: Solve log 2x −1( ) = log 4x − 3( )− log x

log 2x −1( ) = log 4x − 3x

2x −1= 4x − 3x

Property of Logarithms

Multiply both sides by x

Distributive property

(2x - 3)(x - 1) = 0

Check!

Property of Logarithms

x(2x - 1) = 4x - 3

2x2 - x = 4x - 3

2x2 - 5x + 3 = 0 Write quadratic equation in standard form

Solve by factoring

x = 3/2 and x = 1

Day 27

1. Exercises.

Day 27

1. Opener.

Day 30

1. Quiz 4.

1. “Quiz 4”.2. Name.3. Student Number.4. Date.

2. Quiz 4.

1. How long does it take to double an investment of $ 20,000.00 in a bank paying an interest rate of 4% per year compounded monthly?

Find the value of x in the following equations. Check your answers.

2.

3.

4. Character that maintained a robust dispute with Newton over the priority of invention of calculus.

5. How did Evariste Galois die, two days after leaving prison, at age 21?

6. Why isn’t there a Nobel Prize in mathematics?

3x+4 = e5x

log12 x − 7( ) = 1− log12 x − 3( )