8-5 Exponential and Logarithmic Equations Hubarth Algebra II

An equation of the form 𝑏 𝑐𝑥 =𝑎, where the exponent includes a variable, is an exponential Ex. 1 Solving an Exponential Equation Solve 52x = 16. 52x = 16 log 52x = log 16 Take the common logarithm of each side. 2x log 5 = log 16 Use the power property of logarithms. x = Divide each side by 2 log 5. log 16 2 log 5 0.8614 Use a calculator. Check: 52x 16 52(0.8614) 16

Ex. 2 Solving an Exponential Equation by Graphing Solve 43x = 1100 by graphing. Graph the equations y1 = 43x and y2 = 1100. Find the point of intersection. The solution is x 1.684

Ex. 3 Solving an Exponential Equation by Tables Solve 52x = 120 using tables. Enter y1 = 52x – 120. Use tabular zoom-in to find the sign change, as shown at the right. Graph The solution is x  1.487.

Property Change of Base Formula For any positive numbers , 𝑀, 𝑏, and 𝑐, with 𝑏≠1 and 𝑐≠1. 𝑙𝑜𝑔 𝑏 𝑀= 𝑙𝑜𝑔 𝑐 𝑀 𝑙𝑜𝑔 𝑐 𝑏 Ex. 4 Using the Change of Base Formula

Ex. 4 Using the Change of Base Formula Use the Change of Base Formula to evaluate log6 12. Then convert log6 12 to a logarithm in base 3. log6 12 = log 12 log 6 Use the Change of Base Formula. ≈ 1.0792 0.7782 ≈1.387 Use a calculator log6 12 = log3 x Write an equation. 1.387≈ 𝑙𝑜𝑔 3 𝑥 Substitute log6 12 = 1.3868 1.387≈ log 𝑥 log 3 Use the change of base formula 1.387 • log⁡3≈ log⁡𝑥 Multiply each side by log 3. 1.387 • 0.4771≈log⁡𝑥 Use a calculator. The expression log6 12 is approximately equal to 1.3869, or log3 4.589. 0.6617≈log⁡𝑥 Simplify. 𝑥≈ 10 0.6617 Write in exponential form. ≈4.589 Use a calculator.

Ex. 5 Solving a Logarithmic Equation Solve log (2x – 2) = 4. Method 1: log (2x – 2) = 4 2x – 2 = 104 Write in exponential form. 2x – 2 = 10000 x = 5001 Solve for x. Method 2: Graph the equation y1 = log (2x – 2) and y2 = 4. Use Xmin = 4000, Xmax = 6000, Ymin = 3.9 and Ymax = 4.1. Find the point of intersection. The solution is x = 5001.

Ex. 6 Using Logarithmic Properties to Solve an Equation Solve 3 log x – log 2 = 5. 3 log x – log 2 = 5 x3 2 Log ( ) = 5 Write as a single logarithm. x3 2 = 105 Write in exponential form. x3 = 2(100,000) Multiply each side by 2. x = 10 200, or about 58.48. 3

Practice 1. Solve each equation. Round your answer to the nearest ten-thousandth. a. 3 𝑥 =4 b. 6 2𝑥 =21 c. 3 𝑥+4 =101 1.2619 0.8496 0.2009 2. Solve 11 6𝑥 =786 by Graphing Graph 0.4634 3. Evaluate 𝑙𝑜𝑔 5 400 and convert it to a logarithm in base 8 3.7227, 𝑙𝑜𝑔 8 2301 4. Solve log 7−2𝑥 =−1 . Check your answer. 3.45 5. Solve log 6−𝑙𝑜𝑔3𝑥=−2 200