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1. Click (memorize) in a memorizable table below, to begin a flashcard-like process ♦ Try clicking (memorize) in both columns (to decide which side to start with) 2. Guess the answer in your head 3. Click the buttons and follow the directions at the bottom of the table See Help for more detailed directions

Base 2 of n exponent

You may notice a pattern developing in the higher orders. For the higher powers it's more important to notice the millionth and billionth place as opposed to memorizing all the digits

Question (memorize)	Answer (memorize)
20	1
21	2
22	4
23	8
24	16
25	32
26	64
27	128
28	256
29	512
210	1,024
211	2,048
212	4,096
213	8,192
214	16,388
220	1,048,576
230	1,073,741,824
240	1,099,511,627,776

Base b (>2) of n exponent

 

These are common powers you may be expected to perform in your head.

 

Question (memorize)	Answer (memorize)
32	9
33	27
34	81
42	16
43	64
52	25
53	125
62	36
63	216
72	49
73	343
82	64
83	512
92	81
93	729
102	100
103	1,000
106	1,000,000
109	1,000,000,000
1012	1,000,000,000,000

Logorithms

A logarithm of a given number to a given base is the power to which you need to raise the base in order to get the number.

 

In general n = log(x) (given the log's base equals b) because x = b^n and for natural logs, n=ln(x) because x = e^n (the l in ln is lowercase L) We generally assume that if the base isn't specified, the base is 10 (log(100) = 2 because 10^2 = 100)

 

Question (memorize)	Answer (memorize)
log100	2
log1,000,000	6
log(base2)(1,000,000)	~20 (or 19.931...)
e	2.71828...
ln	log with base e
ln(e)	1
ln(e^2)	2
ln(e^3)	3
e^(ln(b))	b
e^(ln(4))	4
e^(p*ln(b))	b^p
e^(2*ln(3))	3^2
ln(ab)	ln(a) + ln(b)
ln(2*3)	ln(2) + ln(3)
ln(c/d)	ln(c) - ln(d)
ln(4/5)	ln(4) - ln(5)
ln(55) = 4.007333...	e^4.007333... ~= 55
(log(10))/(log(e)) = 1/(log(e))	ln(10)
(ln(e))/(ln(10)) = 1/(ln(10))	log(e)

Complex and negative exponents

 

Question (memorize)	Answer (memorize)
e^(-a)	1/(e^a)
e^(-2)	1/(e^2)
e^(1/d)	d'th root of e
e^(1/2)	square root of e
e^(1/3)	third (or cube) root of e
e^(2/5)	fifth root of (e^2)
e^(6/8)	eighth root of (e^6); or fourth root of (e^3)
e^(-2/3)	1/(third root of (e^2))
8^(-2/3)	1/(third root of (8^2)) = 1/4 = 1/(3rd rt of 64) = 1/((3rd rt of 8)^2)
9^(3/2)	(square root of 9)^3 = 27 = 3^3 = square root of (9^3)
(e^a)*(e^b)	e^(a+b)
(e^c)/(e^d)	e^(c-d)
[(e^2)*(e^3)]/(e^4)	e^((2+3)-4) = e
(e^2)/[(e^3)*(e^4)]	e^(2-(3+4)) = e^(-5) = 1/(e^5)

 

 

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