1. Natural Logarithms

2. If y=ex, then ln y =x. The number e≈2.71828.
The function y=ex has an inverse, the natural logarithmic function.
If y=ex, then ln y =x.

3. 8 Simplifying Natural Logarithms Ex. Write as a single natural log.
3 ln 6 - ln 8
ln 63 - ln 8
ln 63
Use the Power Property
Use the Quotient Property
Enter 63/8 into the calculator 8
ln 27

4. Ex. Write as a single natural log. 5ln2-ln4 3lnx+lny 4ln3+4lnx
ln x3y
ln 81x4

5. (3x+5)2=e4 (3x+5)2≈54.6 3x+5≈7.39 3x+5≈-7.39 x≈0.797 or x≈-4.130
Solving a Natural Logarithmic Equation
Solve each equation
ln(3x+5)2=4
Rewrite in exponential form
(3x+5)2=e4
Use a calculator e4
(3x+5)2≈54.6
Take the square root of each side, use ±
3x+5≈ x+5≈-7.39
Use a calculator
x≈0.797 or x≈-4.130
Solve for x.
x+2 3

6. Solve each equation
lnx=0.1
ln(3x-9)=21
ln( )=12
x=1.105 x=
x+2 __ 3 x=
x+2 3

7. Solving an Exponential Equation
Ex. Use natural logs to solve the equations
7e2x+2.5=20
7e2x=17.5
e2x=2.5
2x=ln 2.5
x=.458
Subtract 2.5 from both sides
divide both sides by 7
rewrite in log form
Divide both sides by 2

8. Ex. Use natural logs to solve the equations