Pre-Calculus/Trig 3



Pre-Calculus/Trig 3 Name __________________________________

Solving Exponential and Logarithmic Equations

Solving Exponential Equations

- An exponential equation is an equation in which there is a variable in the exponent.

- We need to get that variable out of the exponent to be able to solve the equation.

Here is what we already know how to do:

Solve the equation [pic]

- We can change to logarithmic form and use a calculator to evaluate.

Here is an alternative way:

- What you do to one side of an equation, you must also do to the other. As long as you do the same thing to both sides, you are okay.

- As a result, we can take the log or natural log (ln) of both sides and use the laws of logarithms to bring the exponent down.

Here is the same problem again, solve the equation [pic]

Guidelines for Solving Exponential Equations

1. If there is only one exponential term…

1. _____________________________________________________________________

2. _____________________________________________________________________

3. _____________________________________________________________________

Ex: [pic]

2. When there is more than exponential term…

Case 1: The bases are the same or can be made into the same base.

1. ______________________________________________________________________________

2. ______________________________________________________________________________

3. ______________________________________________________________________________

Ex: Solve for x.

[pic]

Case 2: The bases are not the same and cannot be made to be the same.

1. ______________________________________________________________________________

2. ______________________________________________________________________________

3. ______________________________________________________________________________

4. ______________________________________________________________________________

5. ______________________________________________________________________________

Ex: Solve for x.

[pic]

Examples:

Solve for x.

1. [pic] 2. [pic]

3. [pic] (calculator)

Solving Logarithmic Equations

- A logarithmic equation is an equation in which a logarithm of the variable occurs.

- We need to get the variable out from the logarithm.

Here is what we already know how to do:

Solve the equation [pic]

Guidelines for Solving Logarithmic Equations

1. If the is just one logarithm …..

1. ______________________________________________________________________________

2. ______________________________________________________________________________

3. ______________________________________________________________________________

4. ______________________________________________________________________________

Ex: Solve for x.

[pic]

2. If there is more than one logarithm…

1. ______________________________________________________________________________

2. ______________________________________________________________________________

______________________________________________________________________________

3. ______________________________________________________________________________

4. ______________________________________________________________________________

5. ______________________________________________________________________________

6. ______________________________________________________________________________

Ex 1: [pic] Ex 2: [pic]

Examples:

Solve for x.

1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic]

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