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**and Logarithmic Equations**

Section 12.6 Exponential and Logarithmic Equations Phong Chau

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Properties Power Rule:

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**Exponential Equations**

Equations with variables in the exponents are called Exponential Equations For simple equation, use the followingprinciple:

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**Solving simple equations**

These exponential equations are simple because we can express both sides of the equations as a power of the same base

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**Solving exponential equation**

Take the natural logarithm on both sides Use Power Rule Divide both sides by ln 2 This is the exact solution Use calculator to find the approximate solution

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**Strategies for Solving Exponential Equations**

Isolate the exponential term Take the natural logarithm on both sides Use the power rule to pull the x out of the exponent Solve the resulting equation Check the answer in the original equation.

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**Example Solve: e1.32t – 2000 =0 Solution We have: e1.32t = 2000**

Take the natural logarithm ln e1.32t = ln 2000 1.32t = ln 2000 t = (ln 2000)/1.32

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**Example Solve: 3 x +1 – 43 = 0 Solution We have 3 x +1 = 43**

log 3 x +1 = log 43 (x +1)log 3 = log 43 Power rule for logs x +1 = log 43/log 3 x = (log 43/log 3) – 1 The solution is (log 43/log 3) – 1, or approximately

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Example

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The above link is a good website to learn new concept. It has many applications.

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World Population The world population in billions at time t, where t = 0 represents the year 2000, is given by: When will the population reach 12 billions?

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**Strategies for solving Logarithmic Equations**

Move all logarithms to left hand side (LHS) Write the LHS as a single logarithm Rewrite the equation in exponential form Solve the resulting equation Check the answer in the original equation.

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**Solution Example Solve: log2(6x + 5) = 4. log2(6x + 5) = 4 6x + 5 = 24**

Check the solution!

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**Example Solve: log x + log (x + 9) = 1. Solution**

log[x(x + 9)] = 1 x(x + 9) = 101 x2 + 9x = 10 x2 + 9x – 10 = 0 (x – 1)(x + 10) = 0 x – 1 = 0 or x + 10 = 0 x = 1 or x = –10

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**Check x = 1: log 1 + log (1 + 9) 0 + log (10) 0 + 1 = 1 x = –10:**

= 1 TRUE x = –10: log (–10) + log (–10 + 9) FALSE The logarithm of a negative number is undefined. The solution is 1.

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**Example Solution Solve: log3(2x + 3) – log3(x – 1) = 2.**

log3[(2x + 3)/(x – 1)] = 2 (2x + 3)/(x – 1) = 32 (2x + 3)/(x – 1) = 9 (2x + 3) = 9(x – 1) 2x + 3 = 9x – 9 x = 12/7 Check the solution!

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Examples

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Examples

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Group Exercise

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