1. Essential Question: What are some of the similarities and differences between natural and common logarithms.

2.  In section 8-2, we talked about the number e ≈ 2.71828 being used as a base for exponents.  The function e x has an inverse, the natural logarithm function  If y = e x, then log e y = x, which is commonly written as ln y = x  The properties of common logarithms apply to natural logarithms as well

3.  Example 1: Simplifying Natural Logarithms ◦ Write 3 ln 6 – ln 8 as a single logarithm  3 ln 6 – ln 8  power rule  simplify  quotient rule  simplify ln 6 3 – ln 8 ln 216 – ln 8 ln 216 / 8 ln 27

4.  Your Turn ◦ Write each expression as a single logarithm  5 ln 2 – ln 4  3 ln x + ln y  ¼ ln 3 + ¼ ln x ln 8 ln x 3 y

5.  You can use the properties of logarithms to solve natural logarithmic equations  Example 3: Solving a Natural Logarithmic Equation ◦ Solve ln (3x + 5) = 4  ln (3x + 5) = 4  Convert to a base of e  Subtract 5 from each side  Divide each side by 3  e 4 = 3x + 5 54.5982 = 3x + 5 49.5982 = 3x 16.5327 = x

6.  Your Turn ◦ Solve each equation  ln x = 0.1  ln (3x – 9) = 21 1.1052 439,605,247.8277

7.  You can use natural logarithms to solve exponential equations  Example 4: Solving an Exponential Equation ◦ Solve 7e 2x + 2.5 = 20  7e 2x + 2.5 = 20Get the e base by itself  Subtract 2.5 from each side  Divide each side by 7  Convert to a ln  Divide both sides by 2  Use a calculator 7e 2x = 17.5 e 2x = 2.5 ln 2.5 = 2x ln 2.5 / 2 = x 0.4581= x

8.  Your Turn ◦ Solve each equation  e x+1 = 30  2.4012 1.6046

9.  Page 472-473 ◦ Problems 1 – 9 & 15 – 27, odds ◦ Show your work ◦ Remember to round all problems to 4 decimal places (if necessary)