solving exponential and logarithmic equations there are lots of them in sec. 3.5a

Solving Exponential Solving Exponential and Logarithmic and Logarithmic

EquationsEquationsThere are lots of them in Sec. There are lots of them in Sec.

3.5a3.5a

But first, a couple of new properties

One-to-One PropertiesOne-to-One Properties

For any exponential function ,xf x b• If ,u vb b then u v

For any logarithmic function log ,bf x x• If log log ,b bu v then u v

For each of the following, find the exact solutionFor each of the following, find the exact solutionalgebraically, and check it by substituting into thealgebraically, and check it by substituting into theoriginal equation.original equation.

320 1 2 5

x 3

1 1

2 4

x

3 21 1

2 2

x

23

x

A One-to

-One Pro

p!!!

A One-to

-One Pro

p!!!

6x The check???The check???

For each of the following, find the exact solutionFor each of the following, find the exact solutionalgebraically, and check it by substituting into thealgebraically, and check it by substituting into theoriginal equation.original equation.

332 1 4 2

x 3

1 1

4 16

x

3 21 1

4 4

x

23

x

A One-to

-One Pro

p!!!

A One-to

-One Pro

p!!!

6x The check???The check???

Whiteboard: For each of the following, find the Whiteboard: For each of the following, find the exact solution algebraically, and check it by exact solution algebraically, and check it by substituting into the original equation.substituting into the original equation.

23 4 96x 24 32x

The check???The check???

22 52 2x

52 2x 5x

Whiteboard: For each of the following, find the Whiteboard: For each of the following, find the exact solution algebraically, and check it by exact solution algebraically, and check it by substituting into the original equation.substituting into the original equation.

2log 5x 52x The check???The check???

32x

4log 1 1x 11 4x

The check???The check???

3x

For each of the following, find the exact solutionFor each of the following, find the exact solutionalgebraically, and check it by substituting into thealgebraically, and check it by substituting into theoriginal equation.original equation.

43 5 15x 45 5x

The check???The check???

14

x 4x

Solve each of the following equations algebraically.Solve each of the following equations algebraically.Obtain a numerical approximation for your solutionObtain a numerical approximation for your solutionand check it by substituting into the originaland check it by substituting into the originalequation.equation.

0.98 1.6x ln 0.98 ln1.6x ln 0.98 ln1.6x

ln1.6

ln 0.98x 23.264

What if we had justWhat if we had justconverted the originalconverted the originalequation into logarithmicequation into logarithmicform???form???

0.98log 1.6

Whiteboard: Solve each of the following equations Whiteboard: Solve each of the following equations algebraically. Obtain a numerical approximation for algebraically. Obtain a numerical approximation for your solution and check it by substituting into the your solution and check it by substituting into the original equation.original equation.

0.04580 240xe 0.045 3xe 0.045ln ln 3xe

0.045 ln ln 3x eln 3

0.045x 24.414

Whiteboard: Solve each of the following equations Whiteboard: Solve each of the following equations algebraically. Obtain a numerical approximation for algebraically. Obtain a numerical approximation for your solution and check it by substituting into the your solution and check it by substituting into the original equation.original equation.

7 3 2xe 3 5xe

5 3xe

0.511 ln 5 3x ln 5 3x

Whiteboard: Solve each of the following equations Whiteboard: Solve each of the following equations algebraically. Obtain a numerical approximation for algebraically. Obtain a numerical approximation for your solution and check it by substituting into the your solution and check it by substituting into the original equation.original equation.

3 log 2 5x log 2 2x

1.9922 10x

22 10x