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Summary

Natural logarithms

In a nutshell

The natural logarithm, $\ln(x)$ , is the logarithm with base $e$ , and therefore the functions $e^x$ and $\ln(x)$ are inverses to each other.

The graph of the equation $y = \ln(x)$

A consequence of the functions $e^x$ and $\ln(x)$ being inverse to each other is that the graphs of the equations $y = e^x$ and $y = \ln(x)$ are reflections of each other about the line $y = x$ . This is because $e^{\ln(x)} = \ln(e^{x}) = x$ , and so if $(a,b) = (a,e^a)$ lies on the graph of the equation $y = e^x$ , then the point $(b, \ln(b)) = (b, \ln(e^a)) = (b,a)$ lies on the graph of the equation $y = \ln(x)$ .

Maths; Exponentials and logarithms; KS5 Year 12; Natural logarithms

As $x$ decreases, $e^x$ tends towards $0$ . Consequently, the graph of the equation $y = \ln(x)$ has the $y$ -axis as an asymptote. This means that $\ln(x)$ is only defined when $x$ is a positive real number.

$e^0 = 1$ , and so $\ln(1) = 0$ . The graph of the equation $y = \ln(x)$ therefore passes through the point $(1,0)$ .

Example 1

Solve the equation $\ln(x-5) + \ln(x+2) = \ln(18)$ 

Use the multiplication law for logarithms to get that $\ln(x-5) + \ln(x+2) = \ln((x-5)(x+2))$ . Expand this, and raise $e$ to the power of both sides of the equation:

$\begin{aligned}\ln((x-5)(x+2)) &= \ln(18)\\\ln(x^2 - 3x - 10) &= \ln(18)\\e^{\ln(x^2 - 3x - 10)} &= e^{\ln(18)}\\x^2 - 3x - 10 &= 18\\x^2 - 3x - 28 &= 0\\(x-7)(x+4) &= 0\end{aligned}$ 

It looks like the solutions to the equation are $x = 7$ and $x = -4$ . However, $\ln(x-5)$ is only defined when $x \gt 5$ , and $\ln(x+2)$ is only defined when $x \gt -2$ . So, $\ln(x-5) + \ln(x+2)$ is only defined when $x \gt 5$ .

Therefore, the only solution to the equation $\ln(x-5) + \ln(x+2) = \ln(18)$ is $\underline{x=7}$ .

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Natural logarithms

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FAQs - Frequently Asked Questions

How are the graphs of e^x and ln(x) related?

The graphs of the equations y=e^x and y=ln(x) are reflections of each other about the line y=x. This is because they are each other's inverse function.

Is the natural logarithm defined for negative values?

The natural logarithm ln(x) is only defined when x is a positive real number.

What is the purpose of the natural logarithm?

The natural logarithm, ln(x), is the logarithm with base e, and therefore the functions e^x and ln(x) are inverses to each other.

The graph of the equation

y = \ln(x)

A consequence of the functions

e^x

and

\ln(x)

being inverse to each other is that the graphs of the equations

y = e^x

and

y = \ln(x)

are reflections of each other about the line

y = x

. This is because

e^{\ln(x)} = \ln(e^{x}) = x

, and so if

(a,b) = (a,e^a)

lies on the graph of the equation

y = e^x

, then the point

(b, \ln(b)) = (b, \ln(e^a)) = (b,a)

lies on the graph of the equation

y = \ln(x)

x

decreases,

e^x

tends towards

0

. Consequently, the graph of the equation

y = \ln(x)

has the

y

-axis as an asymptote. This means that

\ln(x)

is only defined when

x

is a positive real number.

e^0 = 1

, and so

\ln(1) = 0

. The graph of the equation

y = \ln(x)

therefore passes through the point

(1,0)

Example 1

Solve the equation $\ln(x-5) + \ln(x+2) = \ln(18)$ 

Use the multiplication law for logarithms to get that $\ln(x-5) + \ln(x+2) = \ln((x-5)(x+2))$ . Expand this, and raise $e$ to the power of both sides of the equation:

$\begin{aligned}\ln((x-5)(x+2)) &= \ln(18)\\\ln(x^2 - 3x - 10) &= \ln(18)\\e^{\ln(x^2 - 3x - 10)} &= e^{\ln(18)}\\x^2 - 3x - 10 &= 18\\x^2 - 3x - 28 &= 0\\(x-7)(x+4) &= 0\end{aligned}$ 

It looks like the solutions to the equation are $x = 7$ and $x = -4$ . However, $\ln(x-5)$ is only defined when $x \gt 5$ , and

\ln(x+2)

is only defined when

x \gt -2

. So,

\ln(x-5) + \ln(x+2)

is only defined when

x \gt 5

Therefore, the only solution to the equation $\ln(x-5) + \ln(x+2) = \ln(18)$ is $\underline{x=7}$ .

FAQs - Frequently Asked Questions

How are the graphs of e^x and ln(x) related?

The graphs of the equations y=e^x and y=ln(x) are reflections of each other about the line y=x. This is because they are each other's inverse function.

Is the natural logarithm defined for negative values?

The natural logarithm ln(x) is only defined when x is a positive real number.

What is the purpose of the natural logarithm?

The natural logarithm, ln(x), is the logarithm with base e, and therefore the functions e^x and ln(x) are inverses to each other.

Natural logarithms

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Further kinematics

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Summary

Natural logarithms

​​In a nutshell

The graph of the equation y=ln⁡(x)y = \ln(x)y=ln(x)​​

Example 1

Learn with Basics

Unit 1

Exponential functions

Unit 2

Logarithms

Jump Ahead

Unit 3

Natural logarithms

Final Test

FAQs - Frequently Asked Questions

How are the graphs of e^x and ln(x) related?

Is the natural logarithm defined for negative values?

What is the purpose of the natural logarithm?

Natural logarithms

Videos

Summary

Exercises

Select Lesson

Exam Board

Further kinematics

Application of forces

In a nutshell

The graph of the equation $y = \ln(x)$

In a nutshell

The graph of the equation $y = \ln(x)$