Chapter Five - LSUMath



Section 5.5 Exponential and Logarithmic Equations

Objectives

1. Solving Exponential Equations

2. Solving Logarithmic Equations

Objective 1: Solving Exponential Equations

We have already solved exponential equations using the method of relating the bases. But suppose we are given an exponential equation in which the bases cannot be related. We can now use some properties of logarithms to solve these equations. Recall the following logarithmic properties.

If [pic], then [pic]. Logarithm Property of Equality

[pic] Power Rule for Logarithms

When we cannot easily relate the bases of an exponential equation, we will use logarithms and their properties to solve them. The methods used to solve exponential equations are outlined below.

Solving Exponential Equations

• If the equation can be written in the form [pic], then solve the equation[pic].

• If the equation cannot be written in the form [pic]:

1. Use the Logarithm Property of Equality to “take the log

of both sides” (typically using base 10 or base e).

2. Use the Product Rule of Logarithms to “bring down” any exponents.

3. Solve for the given variable.

Objective 2: Solving Logarithmic Equations

In Section 5.4 we learned how to solve certain logarithmic equations by using the logarithm property of equality. That is, if we can write logarithmic equation in the form [pic], then [pic].

Properties of Logarithms

If [pic], [pic]and v represent positive numbers, and r is any real number, then

[pic] Product Rule for Logarithms

[pic] Quotient Rule for Logarithms

[pic] Power Rule for Logarithms

When solving logarithmic equations, it is important to always verify the solutions. Logarithmic equations often lead to extraneous solutions.

When a logarithmic equation cannot be written in the form [pic], we follow the steps outlined below.

Solving Logarithmic Equations

1. Determine the domain of the variable.

2. Use properties of logarithms to combine all logarithms and write as a single logarithm if needed.

3. Eliminate the logarithm by rewriting the equation in exponential form.

4. Solve for the given variable.

5. Check for any extraneous solutions. Verify that each solution is in the domain of the variable.

[pic]

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