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

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

Unit 3: Exponential and Logarithmic Functions Activity 6: Solving Exponential Equations Method 1: Common Base Method Using trial and error, let us find the value of x that makes the LS = RS x 2x-3 8 Does LS=RS? 1 21-3=2-2=1/4 NO 2 ½ 3 4 5 6 YES x 2x-3 8 Does LS=RS? 1 21-3=2-2=1/4 NO 2 3 4 5 6 Work out the table then click ANSWER to see the answer ANSWER The solution is x = 6 Since, 26-3=8

Unit 3: Exponential and Logarithmic Functions Activity 6: Solving Exponential Equations Method 1: Common Base Method How can we solve this without trial and error? 1. Write the left side as a power with base 2 3. Solve for x 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

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

Unit 3: Exponential and Logarithmic Functions Activity 6: Solving Exponential Equations Method 1: Common Base Method Using trial and error, let us find the value of x that makes the LS = RS x 2x+2 4x Does LS=RS? 4 1 NO 8 2 16 YES 3 32 64 x 2x+2 4x Does LS=RS? 1 2 3 Work out the table then click ANSWER to see the answer ANSWER The solution is x = 2 Since, 22+2=24=16=42

Unit 3: Exponential and Logarithmic Functions Activity 6: Solving Exponential Equations Method 1: Common Base Method How can we solve this without trial and error? 1. Write the left side as a power with base 2 3. Solve for x 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

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

Unit 3: Exponential and Logarithmic Functions Activity 6: Solving Exponential Equations Method 1: Common Base Method Example 3 Solve: 2(22x)= 1 1. Simplify equation using exponent laws 3. Simplify algebraically until you have a two power equation LS = RS 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

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

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

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

Unit 3: Exponential and Logarithmic Functions Activity 6: Solving Exponential Equations 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.

Unit 3: Exponential and Logarithmic Functions Activity 6: Solving Exponential Equations 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.

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

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

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