Math Lesson 13.4.4 - Natural Logarithm Rules

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Welcome to our Math lesson on Natural Logarithm Rules, this is the fourth lesson of our suite of math lessons covering the topic of Natural Logarithm Function and Its Graph, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Natural Logarithm Rules

The rules for natural logarithms are similar to those of standard logarithms given the fact that natural logarithms are a special type of logarithm. Anyway, let's briefly write the natural logarithm rules, as we will need them when dealing with logarithmic functions.

1. Product rule

For example,

Indeed, the calculator gives

Thus,

2. Quotient rule

For example,

Indeed, the calculator gives

Thus,

3. Power rule

For example,

Indeed, the calculator gives

Thus,

4. Euler's Number raised to ln rule

For example,

Indeed, the calculator gives

Thus,

Oher natural logarithm rules

There are some other rules of natural logarithms not specified into any category. They are:

1. ln 1 = 0
   Indeed, it is known that any number raised to the zeroth power gives 1. Euler's Number e makes no exception in this regard. Thus, since ln 1 = log_e 1, we have
   ln 1 = log_e⁡1 = 0 as e⁰ = 1
2. ln 0 is undefined (it is minus infinity)
   Indeed, the only exponent to give a zero power when a positive number is raised to that power is the minus infinity. In the specific case, we have
   ln 0 = -∞ as e^-∞ = 0
3. The ln of negative numbers is undefined except for -1, for which we have the Euler's identity
   ln (-1) = i ∙ π
   where 'i' is the imaginary number, which we will explain in the upcoming chapters of this course. For now, we can say only that
   i = √(-1)

Example 1

Write the following expressions in the simplest form and then calculate their value if possible.

1. ln 24-3 ln 2-ln 3
2. ln x/3 - ln x² + ln (4x) - 2 ln 2
3. ln 15 + ln (3e) + ln e/5 - 3 ln 21 - ln 243

Solution 1

1. Applying the properties of natural logarithm yields
   ln 24 - 3 ln 2 - ln 3
   = ln (8 ∙ 3) - 3 ln 2 - ln 3
   = ln 8 + ln 3 - 3 ln 2 - ln 3
   = ln 2³ + ln 3 - 3 ln 2 - ln 3
   = 3 ln 2 + ln 3 - 3 ln 2 - ln 3
   = (3 ln 2 - 3 ln 2 ) + (ln 3 - ln 3 )
   = 0 + 0
   = 0
2. Again, applying the properties of natural logarithm yields
   ln x/3 - ln x² + ln (4x) - 2 ln 2
   = ln x - ln 3 - 2 ln x + ln 4 + ln x - 2 ln 2
   = ln x - ln 3 - 2 ln x + ln 2² + ln x - 2 ln 2
   = ln x - ln 3 - 2 ln x + 2 ln 2 + ln x - 2 ln 2
   = (ln x - 2 ln x + ln x ) - ln 3 + (2 ln 2 - 2 ln 2 )
   = 0 - ln 3 + 0
   = ln 3
   = 1.0986
3. Still applying the properties of natural logarithms yields
   ln 15 + ln (3e) + ln e/5 - 3 ln 21 - ln 243
   = ln (5 ∙ 3) + ln (3 ∙ e) + ln e/5 - 3 ln (7 ∙ 3) - ln 243
   = ln 5 + ln 3 + ln 3 + ln e + ln e - ln 5 + 3 ln 7 + 3 ln 3 - ln 7³
   = ln 5 + ln 3 + ln 3 + ln e + ln e - ln 5 + 3 ln 7 + 3 ln 3 - 3 ln 7
   = ln 5 + ln 3 + ln 3 + ln e + ln e - ln 5 + 3 ln 7 + 3 ln 3 - 3 ln 7
   = ln 5 + ln 3 + ln 3 + ln e + ln e - ln 5 + 3 ln 7 + 3 ln 3 - 3 ln 7
   = (3 ln 7 - 3 ln 7 ) + (ln 5 - ln 5 ) + (ln 3 + ln 3 + 3 ln 3 ) + (ln e + ln e )
   = 0 + 0 + 5 ln 3 + 2 ln e
   = 5 ln 3 + 2 ∙ 1
   = 5 ∙ 1.0986 + 2
   = 5.493 + 2
   = 7.493

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