Natural Log Formula, Definition, Solved Examples 

The Natural Log Formula Logarithms present an alternate representation of exponents, Comprehending logarithms isn't overly complex; it primarily involves recognizing that a logarithmic expression serves as an alternative to an exponential one. Let's articulate the natural log formula in terms of exponents: If e ^x =a, then log ^e ​ _a =x, and the converse holds true. Interpreting log ^e ​ _a =x can be verbalized as 'The logarithm of a with a base of e equals x.' Logarithms with the base e are termed natural logarithms.

What is the Natural Log Formula

Let's explore four natural log formulas that elucidate different properties of logarithms:

Product Rule: When dealing with the logarithm of a product of two numbers, it translates to the sum of the logarithms of the individual numbers.

log _e ​ (m⋅n)=log _e ​ (m)+log _e ​ (n)

Quotient Rule: The logarithm of a quotient of two numbers equates to the difference between the logarithms of the individual numbers.

log _e ​ ( n m ​ )=log _e ​ (m)−log _e ​ (n)

Power Rule: The exponent of the argument within a logarithm can be moved in front of the logarithm.

log _e ​ (a m )=m⋅log _e ​ (a)

Change of Base Rule: This rule allows the alteration of the base of a logarithm using either of these expressions:

log _e ​ (a)= log _c ​ (e) log _c ​ (a) ​

or

log _e ​ (a)⋅log _c ​ (e)=log _c ​ (a)

Natural Log Formula Solved Examples

Example 1: Determine the integer value of  x using natural log formulas from the equation e ^3x =9.

Solution: e ^3x =9

ln(e 3x )=ln(9)

3x=ln(9)

x= ln(9) ​/ 3

Example 2: Simplify the expression: 6ln _e (y)+2ln _e (4y)−ln _e (8y ⁶ ) Solution:

6ln _e (y)+2ln _e (4y)−ln _e (8y ⁶ )

=ln _e (y ⁶ )+ln _e ((4y) ² )−ln _e (8y ⁶ )

= ln _e (y ⁶ )+ln _e (16y ² )−ln _e (8y ⁶ )

= lne(y ⁶ ⋅16y ² )−lne(8y ⁶ )

=lne(16y ⁸ )−lne(8y ⁶ )

=lne(16y ⁸ /8y ⁶ )

=lne(2y ² )

Example 3: To solve for  x in the equation e ^2x =20

We start by taking the natural logarithm of both sides to isolate the variable x.

So, we apply the natural logarithm function (ln) to both sides of the equation: ln(e ^2x )=ln(20)

The property of logarithms allows us to bring down the exponent as a multiplier: 2x⋅ln(e)=ln(20)

Since the natural logarithm of the base  e is 1, we're left with:

2x=ln(20)

Finally, to solve for  x, we divide both sides by 2:

x= ln(20) / 2 ​

Example 4: In the expression 4lne(x)−lne(x ² ), we want to simplify it by using the properties of logarithms.

We rewrite the expression using the properties of logarithms:

4lne(x)−lne(x ² )

=lne(x ⁴ )−lne(x ² )

=lne(x ⁴ /x ² )

= lne(x ² )

Example 5: Given the equations e ^x =7 and e ^2x =49, we're tasked with finding the value of  x.

From the first equation e ^x =7, we know that x=ln(7).

Using the second equation e ^2x =49,

we substitute the value of  x obtained earlier: e ^2x =49

e ^2ln(7) =49

(e ^ln(7) ) ² =49

7 ² =49

This shows that the value of  x is indeed ln(7), and by extension, the value of  x is 2, since e ^2x =49 holds true when x=2.

These step-by-step explanations illustrate how natural logarithm formulas are applied to solve equations and simplify expressions involving logarithmic functions.

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