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Unit 3: Exponential and Logarithmic Functions

Learn how to solve exponential equations using common base and logarithmic methods with examples and step-by-step solutions in this activity.

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Unit 3: Exponential and Logarithmic Functions

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  1. Activity 6: Solving Exponential Equations Unit 3: Exponential and Logarithmic Functions In this activity you will learn how to solve exponential equations using two methods: the common base method and the logarithmic method.

  2. Activity 6: Solving Exponential Equations Unit 3: Exponential and Logarithmic Functions Example 1 Solve for x: 2x-3 = 8 Method 1: Common Base Method

  3. Activity 6: Solving Exponential Equations Unit 3: Exponential and Logarithmic Functions Using trial and error, let us find the value of x that makes the LS = RS Work out the table then click ANSWER to see the answer Method 1: Common Base Method ANSWER The solution is x = 6 Since, 26-3=8

  4. Activity 6: Solving Exponential Equations Unit 3: Exponential and Logarithmic Functions How can we solve this without trial and error? 1. Write the left side as a power with base 2 3. Solve for x Method 1: Common Base Method 2x-3=23 x = 6 2. Set the exponents as an equation 4. What do you notice? You get the same solution as trial and error x – 3 = 3

  5. Activity 6: Solving Exponential Equations Unit 3: Exponential and Logarithmic Functions Example 2 Solve for x: 2x+2 = 4x Method 1: Common Base Method

  6. Activity 6: Solving Exponential Equations Unit 3: Exponential and Logarithmic Functions Using trial and error, let us find the value of x that makes the LS = RS Work out the table then click ANSWER to see the answer Method 1: Common Base Method ANSWER The solution is x = 2 Since, 22+2=24=16=42

  7. Activity 6: Solving Exponential Equations Unit 3: Exponential and Logarithmic Functions How can we solve this without trial and error? 1. Write the left side as a power with base 2 3. Solve for x Method 1: Common Base Method 2x+2=(22)x =22x x = 2 2. Set the exponents as an equation 4. What do you notice? You get the same solution as trial and error x + 2 = 2x

  8. Activity 6: Solving Exponential Equations Unit 3: Exponential and Logarithmic Functions Process 1. Simplify equation using exponent laws 2. Write all powers with the same base Method 1: Common Base Method 3. Simplify algebraically until you have a two power equation LS = RS 4. Set your exponents equal and solve

  9. Activity 6: Solving Exponential Equations Unit 3: Exponential and Logarithmic Functions Example 3 Solve: 2(22x)= 1 1. Simplify equation using exponent laws 3. Simplify algebraically until you have a two power equation LS = RS Method 1: Common Base Method 2(22x)= 1 22x+1 = 1 22x+1 = 20 4. Set your exponents equal and solve 2. Write all powers with the same base 2x+ 1 = 0 2x = -1 x = -1/2 22x+1 = 20

  10. Activity 6: Solving Exponential Equations Unit 3: Exponential and Logarithmic Functions Example 3 Solve: 2(22x)= 1 You can check the solution using LS=RS. Below is a graphical check of the solution Method 1: Common Base Method y=2(22x) x=1/2 y=1

  11. Activity 6: Solving Exponential Equations Unit 3: Exponential and Logarithmic Functions Example 4 Solve: 27x(92x-1) = 3x+4 33x(32)2x-1 = 3x+4 Method 1: Common Base Method 33x(34x-2) = 3x+4 37x-2 = 3x+4 7x – 2 = x + 4 x = 1

  12. Activity 6: Solving Exponential Equations Unit 3: Exponential and Logarithmic Functions Example 5 Solve: 4x+3 + 4x = 1040 4x(43) + 4x = 1040 Method 1: Common Base Method 64(4x)+ 4x = 1040 65(4x)= 1040 (4x)= 16 (4x)= 42 .:x = 2

  13. Activity 6: Solving Exponential Equations Unit 3: Exponential and Logarithmic Functions Method 2: Solving with Logarithms Example 6 Solve for x: 5x = 27 Looking at this equation we can see that 27 cannot be made as a power with a base of 5 Let us estimate the value of x 52 = 25 53 = 125 x must be between 2 and 3 but closer to 2.

  14. Activity 6: Solving Exponential Equations Unit 3: Exponential and Logarithmic Functions Method 2: Solving with Logarithms Example 6 Solve for x: 5x = 27 We cannot solve for x since it is an exponent. How can we bring the exponent down so we can solve for it? We can use the logarithmic power law: logbmn = nlogbm on the equation.

  15. Activity 6: Solving Exponential Equations Unit 3: Exponential and Logarithmic Functions Method 2: Solving with Logarithms Example 6 Solve for x: 5x = 27 log5x = 27 log5x = log27 log5x = log27 xlog5 = log27 log5x = log27 xlog5 = log27 log5 log 5 log5x = log27 xlog5 = log27 log5 log 5 x = 2.048 Remember, whatever is done on one side must be done to the other Use the Laws of Logarithms to bring down the exponent Solve the equation by isolating the variable Use your calculator to determine log25/log5 Try another example

  16. Activity 6: Solving Exponential Equations Unit 3: Exponential and Logarithmic Functions Method 2: Solving with Logarithms Example 7 Solve for x: 4x = 8(x+3) log24x = log28(x+3) log24x = log28(x+3) xlog24 = (x+3)log28 log24x = log28(x+3) xlog24 = (x+3)log28 xlog222 = (x+3)log223 log24x = log28(x+3) xlog24 = (x+3)log28 xlog222 = (x+3)log223 x(2) = (x+3)(3) log24x = log28(x+3) xlog24 = (x+3)log28 xlog222 = (x+3)log223 2x = 3(x+3) log24x = log28(x+3) xlog24 = (x+3)log28 xlog222 = (x+3)log223 2x = 3(x+3) 2x = 3x+9 log24x = log28(x+3) xlog24 = (x+3)log28 xlog222 = (x+3)log223 2x = 3(x+3) x = -9 Set the log to both sides. Use a base of 2 for both sides. Use the Power law of logarithms to bring down the exponents Write 4 and 8 as powers of base 2 Using the power property of logarithms the following occurs Logbbm=m Solve the equation by isolating the variable

  17. Activity 6: Solving Exponential Equations Unit 3: Exponential and Logarithmic Functions Completed Activity! Go back to the activity home page and start working on the assignment for this activity

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