 New Zealand
Level 8 - NCEA Level 3

Natural Logarithms

Interactive practice questions

The natural base $e$e (Euler’s number) is defined as $e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n$e=limn(1+1n)n

The table shows the value of $\left(1+\frac{1}{n}\right)^n$(1+1n)n using various values of $n$n.

$n$n $\left(1+\frac{1}{n}\right)^n$(1+1n)n
$1$1 $2$2
$100$100 $1.01^{100}=2.704813$1.01100=2.704813 ...
$1000$1000 $1.001^{1000}=2.716923$1.0011000=2.716923 ...
$10000$10000 $1.0001^{10000}=2.718145$1.000110000=2.718145 ...
$100000$100000 $1.00001^{100000}=2.718268$1.00001100000=2.718268 ...
a

Evaluate $\left(1+\frac{1}{n}\right)^n$(1+1n)n for $n=1000000$n=1000000, correct to six decimal places.

b

Which of the following is the closest approximation of $e$e?

$2.718280821$2.718280821

A

$2.718281828$2.718281828

B

$2.718281820$2.718281820

C

$2.718281818$2.718281818

D

$2.718280821$2.718280821

A

$2.718281828$2.718281828

B

$2.718281820$2.718281820

C

$2.718281818$2.718281818

D
Easy
Approx 3 minutes

Which of the following are the correct logarithmic forms of the equation $e^x=3$ex=3? Choose all the correct options.

State the value of $\ln e$lne.

Find the value of $5\log_ee$5logee.

Outcomes

M8-7

Form and use trigonometric, polynomial, and other non-linear equations

91575

Apply trigonometric methods in solving problems