Exponent, Exponential and Logarithmic Equations



Exponent, Exponential and Logarithmic Equations

Solving Exponent Equations:

Ex: [pic]

1. multiply the exponent to its reciprocal (signs stay the same); what you do to one side of the equation you must do to the other side of the equation

[pic]

2. solve for x by plugging the values in the calculator

x = 8 ٨ (1/3)

x = 2

Solving Exponential Equations:

Ex: [pic]

1. If need, make the bases the same

[pic] [pic]

[pic]

2. Multiply the exponents if needed

[pic]

3. Ignore bases and set exponents equal to each other

[pic]

4. Solve for x:

[pic]

-2x -2x

2 = x

Rewriting exponents into logarithms:

LOG = EXPONENT

Ex: [pic]

1. Your exponent is your answer to the log problem.

Log = 2

2. The base is still the base just gets smaller.

[pic]

3. Whatever you didn’t use goes in the missing spot.

[pic]

Rewriting logarithms into exponents:

LOG = EXPONENT

Ex: [pic]

1. Whatever the log equals is your exponent.

Exponent is 2.

2. The base stays the same, just gets bigger.

Base is 2.

[pic]

3. Whatever you didn’t use is your answer.

[pic]

Solving logarithms:

Ex: [pic]

1. Rewrite log problem as an exponent problem

[pic]

2. Solve for x:

[pic]

[pic]

x = 8 ٨ (1/3)

x = 2

Logarithmic Rules:

Product Rule: [pic]

Quotient Rule: [pic]

Power Rule: [pic]

Solving logarithmic equations:

Ex: [pic]

1. Solve each log individually

[pic] [pic]

[pic] [pic]

[pic] [pic]

2. Replace logs with their values

[pic]

[pic]

3. Isolate the variable with the log

[pic]

-4 -4

[pic]

4. Solve log problem

[pic]

[pic]

Solving more complex ones:

Ex: [pic]

1. use any log laws possible

[pic]

2. Rewrite log into an exponent problem

[pic]

3. Solve for x:

[pic]

[pic]

(x – 5)(x + 2) = 0

x = 5, -2

4. Reject any negative answers

x = 5

Solving Exponential Equations where the bases cannot be the same:

Ex: [pic]

1. log both sides with the base of 10

[pic]

2. use any log laws needed

[pic]

3. solve for x:

[pic]

[pic]

[pic]

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