Examples Using Natural Log Formula
Example 1: Solve,
- ex = 10
- ln x = 2
- eln 15
- ln(e29)
- ln(39)
- ln(15/4)
- ln(39)
- log57
Solution:
- ex = 10
x = ln 10
x = 2.303
- ln x = 2
x = e2
x = 7.389
- eln 15 = 15
- ln(e29) = 29
- ln (39)
= ln(13 Γ 3)
= ln 13 + ln 3
= 2.565 + 1.099 = 3.664
- ln (15/4)
= ln 15 β ln 4
= 2.708 β 1.386 = 1.322
- ln(39)
= 9 Γ ln 3
= 9 Γ 1.099 = 9.891
- log5 7
= (ln 7)/(ln 5)
= 1.946/1.609 = 1.209
Example 2: Solve, ln(15x β 3) = 2
Solution:
ln(15x β 3) = 2
15x β 3 = e2
15x -3 = 7.389
15x = 10.389
x = 10.389/15 β x = 0.6926
Example 3: If 8exy + 2 = 98 and 2ez + 3 = 79, then find the value of x + y, where z = x2 + y2
Solution:
8exy + 2 = 98
8exy = 98-2 = 96
exy = 96/8 = 12
ln(exy) = ln 12
xy = 2.4849β¦(i)
2ez + 3 = 79
2ez = 79-3 = 76
ez = 76/2 = 38
ln(ez) = ln 38
z = 3.6375β¦(ii)
z = x2 + y2
Now, (x + y)2 = x2 + y2 + 2xy
From eq(i) and eq(ii)
(x + y)2 = 3.6375 + 2 Γ 2.4849
(x + y)2 = 3.6375 + 4.9698
(x + y)2 = 8.6073
(x + y) = β8.6073
x + y = 2.933
Example 4: Simplify y = ln 25 β ln 15
Solution:
y = ln 25 β ln 15
y = ln(5 Γ 5) β ln(5 Γ 3)
y = ln 5 + ln 5 β [ln 5 + ln3]
y = ln 5 + ln 5 β ln 5 β ln3
y = ln 5 β ln 3
y = ln (5/3)
y = 0.511
Example 5: Solve: ln(e15) + e 2+x = 16
Solution:
ln(e15) + e 2+x = 16
β 15 + e2+x = 16
β e2+x = 16 β 15
β e2+x = 1
β ln( e2+x )= ln 1
β 2 + x = 0
β x = -2
Example 6: Evaluate: p = log35 β log36 + log310
Solution:
Using base change of log formula,
p = (ln 5/ ln 3) β (ln 6/ ln 3) + (ln 10/ ln 3)
p = [ln 5 -(ln 6 + ln 10)] / ln 3
p = [ln 5 β ln (6 Γ 10)]/ ln 3
p = [ln 5 β ln 60]/ ln 3
p = [ln(5/60)] / ln 3
p = [ln(1/12)] / ln 3
p = [ln (12)-1] / ln 3
p = [-1Γln 12] / ln 3
p = -ln 12 / ln 3
p = -2.262
Natural Log
Natural Log in mathematics is a way of representing the exponents. We know that a logarithm is always defined with abase and for the natural log, the base is βeβ. The natural log is used for solving various problems of Integration, Differentiation, and others.
In this article, we will learn about Natural log, Natural Log Formula, Examples, and others in detail.
Table of Content
- What is Natural Log?
- Natural Log Formula
- Natural Logarithms Table
- Natural Log Derivatrive
- Natural Log Integration
- Natural Lag Laws