8.5 – Exponential and Logarithmic Equations

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Presentation transcript:

8.5 – Exponential and Logarithmic Equations

CHANGE OF BASE FORMULA where M, b, and c are positive numbers and b, c do not equal one. Ex: Rewrite log515 using the change of base formula

Steps for solving exponential equations Take a common logarithm of each side Use the power property of logarithms Solve for x by dividing Use a calculator to find the approximate value

Solving Exponential Equations Solve . Round to the nearest ten-thousandth. 1. Take the log of both sides 2. Use the power property 3. Solve for x. X=1.2619 4. Use a calculator. Check your answer – 31.2619=4

Another Example Solve . Round to the nearest ten-thousandth. 1. Take the log of both sides 2. Use the power property 3. Solve for x. X=4.2009 – 4 = 0.2009 4. Use a calculator. Check your answer – 30.2009+4=101

Let’s try some

Let’s try some

Let’s try some

Let’s try some

CHANGE OF BASE – HOW IT WORKS Use the change of base formula to evaluate . Then convert it to a logarithm of base 2. 1. Rewrite using the change of base formula 2. Use a calculator 3. Write an equation to convert to base 2

CHANGE OF BASE – HOW IT WORKS 4. Substitute log315=2.4650 5. Rewrite using the change of base formula 6. Multiply both sides of the equation by log2 7. Use a calculator; simplify. 8. Write in exponential form. X=5.5208 9. Use a calculator. Log315 is approximately equal to 2.4650 or log25.5208

Let’s try one Use the change of base formula to evaluate . Then convert it to a logarithm of base 8. 1. Rewrite using the change of base formula 2. Use a calculator 3. Write an equation to convert to base 2

4. Substitute log5400=3.727 5. Rewrite using the change of base formula 6. Multiply both sides of the equation by log8 7. Use a calculator; simplify. 8. Write in exponential form. X=2301 9. Use a calculator. Log5400 is approximately equal to 3.7227 or log82301

SOLVING SIMPLE LOG EQUATIONS 1. Use the product property 2. Write in exponential form. 3. Simplify 4. Solve for x.

Let’s try some

Let’s try some

Let’s try some

Let’s try some

Solving exponential equations with a graphing calculator Type two equations into y= 2. Graph. Suggest Zoom fit (0) especially for large values 3. Use the calc function to find the intersection of the two graphs. Solution: 2.0408