Download 7.6 Notes Key

Fisher, Chapter 7 Section 6
Algebra 2
Name:______________________________
7.6 Notes
Date:_______________Pd:_________________
7.6 – Natural Logarithms
I. Natural Logarithms
ln_____
Loge y = x
Natural Logarithmic Function: ____
same thing as
equals
ln y =x
exponential function base e______________
What is 𝑒? _____
Recall:
Notation for Natural Logarithm:
Common Logarithm
log is called the common log
*The base is 10
Natural Logarithm
ln is called the natural log
*The base is e
log x  log 10 x
ln x  log e x
To solve a natural logarithmic equation, rewrite it in exponential form and/or use properties of
logarithims.
II. Simplifying a Natural Logarithm
**The rules for simplifying logarithms also apply to natural logarithms.
Example #1:
Get this #
from calc
ln (2x-7)² =2
loge (2x-7)² =2
e² =(2x-7)²
Step #1: Rewrite in exponential form
7.39=(2x-7)²
√7.39 = √(2x − 7)²
±2.72 = 2x-7
2.72+7=2x
x=4.82
or -2.72 +7 = 2x
x=2.14
1
Fisher, Chapter 7 Section 6
Write each as a single natural logarithm.
Example #1:
ln 6- ln 5
= ln (6/5)
= 1.11
4.
1
ln x  ln y  ln 5
2
2. ln 4  2 ln 5
3. ln x  2 ln 2x
5. 5 ln m  3 ln n
6. 2 ln 8  3 ln 4
IV. Solving a Natural Logarithmic Equation. Look back in your notes!!
**The rules for solving logarithmic equations also apply to natural logarithms.
Solve each equation.
2
1.
ln (2x-1) = 4
ln √(2x − 1)² =√4
ln(2x-1)=2
ln2= 2x-1
e2=2x-1
7.38=2x-1
+1 +1
8.38= 2x
2
2
x = 4.19
2. ln x =2
ln e =1
3. ln( x  5) 2  4
2
Fisher, Chapter 7 Section 6
Solve each equation.
4. ln 2x  ln 3  2
5. ln x  ln x  1  ln 6
6. 2 ln 2x  1
III. Solving an Exponential Equation containing e.
**The rules for solving exponential equations also apply to equations containing 𝑒. However, instead of taking
the common logarithm of both sides, we will use the _____(ln) natural___logarithm
ln ea = a
e ln a = a
Example #1:
e 3x = 7
ln 7 = 3x
3 =x
Solve each equation.
1.
ex = 15
lne=1
loge X =lnx
Use your Properties of Exponent
or
ln e 3x =ln 7
3xlne = ln 7
3x= ln7
3
3
2. e3x
+ 5 = 30
3. 2e x 1  20
2.71
3