Section 11-6
Natural Logarithms

Often when you solve problems involving the
number e, it is best to use logarithms to the
base e. These are called natural logarithms and
are written ln x
Also like logarithms, sometimes you know the
natural log of a number x and must find x. The
antilogarithm of a natural logarithm is written
antiln x.


Convert log 291 to a natural logarithm and
evaluate.
1. Log 5 291 = log e 291 / log e 5 =
ln 291 / ln 5 = 3.52503
5
2. Solve 7.2 = -28.8 ln x
-.25714 = ln x
antiln (-.25714) = antiln ln x
.78 ≈ x


Solve each equation or inequality by using
natural logarithms.
x 1
2x
1. 5 = 7
2x ln 5 = x+1 ln 7
2x (1.60944) = x+1 (1.94591)
3.21888x=1.94591x+1.94591
1.27297x=1.94591
2
x= 1.5287
1.386x - 1.386 < 2.49
2. 4
x 2 1
< 12
x 2  1 ln 4 < ln 12
x 2  1 (1.38629) < 2.48491
2
1.386x < 3.876
2
x < 2.79654
-1.67< x < 1.67

We also use logarithms to measure the
intensity of earthquakes, to determine the
Richter scale of earthquakes, to measure acidity
and the ph scale
 Section
11-6
 Pp. 736-737 #19-39 odds, 55,63