Essential Question: What are some of the similarities and differences between

natural and common logarithms.

In section 8-2, we talked about the number e ≈ 2.71828 being used as a base for exponents.

The function ex has an inverse, the natural logarithm function

If y = ex, then loge y = x, which is commonly written as ln y = x

The properties of common logarithms apply to natural logarithms as well

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 63 – ln 8ln 216 – ln 8ln 216/8

ln 27

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 x3y

4ln 3x

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

e4 = 3x + 554.5982 = 3x + 549.5982 = 3x16.5327 = x

Your Turn◦ Solve each equation

ln x = 0.1

ln (3x – 9) = 21

1.1052

439,605,247.8277

You can use natural logarithms to solve exponential equations

Example 4: Solving an Exponential Equation◦ Solve 7e2x + 2.5 = 20

7e2x + 2.5 = 20 Get 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

7e2x = 17.5e2x = 2.5

ln 2.5 = 2xln 2.5/2 = x

0.4581= x

Your Turn◦ Solve each equation

ex+1 = 30

2.4012

1.604625 7.2 9.1x

e

Page 472-473◦ Problems 1 – 9 & 15 – 27, odds◦ Show your work◦ Remember to round all problems to 4 decimal

places (if necessary)