Natural Logarithm Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Natural Logarithm.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The logarithm with base $e \approx 2.71828$ : $\ln x = \log_e x$ . It is the inverse function of $e^x$ .

If $e^x$ asks 'what do I get after growing continuously for time $x$ ?', then $\ln x$ asks 'how long do I need to grow continuously to reach $x$ ?' The natural log measures time in the world of continuous growth.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: $\ln x$ asks how long continuous growth takes to reach $x$ , and it undoes $e^x$ .

Common stuck point: The procedure for natural logarithm is the easy part; the trap is treating $\ln$ as base 10. Asking "Is the base $e$ (continuous growth), so the inverse I want is $\ln$ rather than a base-10 log?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the base $e$ (continuous growth), so the inverse I want is $\ln$ rather than a base-10 log?

Worked Examples

Example 1

easy

Evaluate

\ln(e^5)

Answer

5

First step

Recall that

\ln(x) = \log_e(x)

, so

\ln

and

e^x

are inverse functions.

Full solution

2
By the inverse property: $\ln(e^a) = a$ for any real number $a$ .
3
Therefore $\ln(e^5) = 5$ .

The natural logarithm

\ln

is the inverse of the exponential function

e^x

. This means

\ln(e^a) = a

and

e^{\ln a} = a

. These inverse relationships are fundamental to working with exponential and logarithmic expressions.

Example 2

medium

Simplify

\ln(x^3) - 2\ln(x) + \ln(e)

Example 3

medium

Solve

\ln(x - 1) + \ln(x + 1) = \ln 8

for

x

Example 4

medium

Find the domain of

f(x) = \ln(x^2 - 4)

Example 5

medium

A culture of bacteria doubles every

3

hours and starts at

200

. Solve for the time when it reaches

1600

, modeling with

N(t) = 200\,e^{kt}

Example 6

hard

Solve

\ln(x) + \ln(x - 3) = \ln(2x + 8)

for

x

Example 7

hard

Carbon-14 has half-life

5730

years. A sample contains

30\%

of its original C-14. Find its age using

N = N_0 e^{-kt}

Example 8

hard

Use

\ln

to solve

5^x = 3^{x+2}

Example 9

challenge

Prove that for

x > 0

\ln x \le x - 1

, with equality only at

x = 1

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

Solve

\ln(2x + 1) = 3

for

x

Example 2

hard

Find the derivative of

f(x) = \ln(x^2 + 1)

and determine where

f

is increasing.

Example 3

easy

Evaluate

\ln 1

Example 4

easy

Evaluate

\ln e

Example 5

easy

Evaluate

\ln(e^5)

Example 6

easy

Use a log property to expand

\ln(xy)

Example 7

easy

Use a log property to expand

\ln\left(\frac{x}{y}\right)

Example 8

easy

Use a log property to rewrite

\ln(x^3)

Example 9

easy

For what values of

x

\ln x

defined (over the reals)?

Example 10

easy

\ln(x + y) = \ln x + \ln y

valid in general? Answer yes or no.

Example 11

medium

Solve

\ln x = 3

for

x

Example 12

medium

Solve

e^{2x} = 7

for

x

Example 13

medium

Write

2\ln x + \ln y

as a single logarithm.

Example 14

medium

Simplify

\ln(e^3) - \ln(e)

Example 15

medium

Solve

\ln(x) + \ln(x - 3) = \ln 10

for

x

Example 16

medium

\ln 2 \approx 0.693

, estimate

\ln 8

Example 17

medium

Expand

\ln\left(\frac{x^2}{y^3}\right)

fully.

Example 18

medium

Solve

\ln(2x) = 4

for

x

Example 19

medium

Simplify

e^{\ln 5 + \ln 2}

Example 20

challenge

Solve

e^{2x} - 5e^x + 6 = 0

for all real

x

Example 21

challenge

Given

\ln a = 2

and

\ln b = 5

, find

\ln\left(\frac{a^3}{\sqrt{b}}\right)

Example 22

challenge

A population grows as

P(t) = 100 e^{0.04t}

. How long until it doubles to 200? (Exact form.)

Example 23

easy

Evaluate

e^{\ln 4}

Example 24

easy

Rewrite

\ln\sqrt{x}

using a logarithm property.

Example 25

easy

Solve

e^x = 10

for

x

Example 26

easy

Expand

\ln(2x^3)

Example 27

easy

Condense

2\ln x + \ln 3

into a single logarithm.

Example 28

medium

Solve

\ln x = 2

for

x

Example 29

medium

Solve

e^{2x} = 7

for

x

Example 30

medium

Simplify

\ln(e^3 x^2) - \ln x

Example 31

medium

Solve

3\ln x = \ln 64

for

x

Example 32

medium

Solve

e^{x+1} = 3

for

x

Example 33

medium

\ln a = 2

and

\ln b = 3

, find

\ln(a^2 b)

Example 34

hard

Find

\dfrac{d}{dx}[\ln(3x^2 + 1)]

Example 35

hard

Solve

\ln(x^2 + 1) = 1 + \ln(x)

for

x > 0

Example 36

hard

Solve

2\ln x - \ln(x + 6) = \ln 4

for

x

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Related Concepts

Logarithm Euler's Number Change of Base Formula Solving Exponential Equations

Background Knowledge

These ideas may be useful before you work through the harder examples.

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