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# Base b of n exponent

rename
Updated
2007-11-12 00:45

## 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 digitsQuestion | Answer |
---|---|

2^{0} | 1 |

2^{1} | 2 |

2^{2} | 4 |

2^{3} | 8 |

2^{4} | 16 |

2^{5} | 32 |

2^{6} | 64 |

2^{7} | 128 |

2^{8} | 256 |

2^{9} | 512 |

2^{10} | 1,024 |

2^{11} | 2,048 |

2^{12} | 4,096 |

2^{13} | 8,192 |

2^{14} | 16,388 |

2^{20} | 1,048,576 |

2^{30} | 1,073,741,824 |

2^{40} | 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 | Answer |
---|---|

3^{2} | 9 |

3^{3} | 27 |

3^{4} | 81 |

4^{2} | 16 |

4^{3} | 64 |

5^{2} | 25 |

5^{3} | 125 |

6^{2} | 36 |

6^{3} | 216 |

7^{2} | 49 |

7^{3} | 343 |

8^{2} | 64 |

8^{3} | 512 |

9^{2} | 81 |

9^{3} | 729 |

10^{2} | 100 |

10^{3} | 1,000 |

10^{6} | 1,000,000 |

10^{9} | 1,000,000,000 |

10^{12} | 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 | Answer |
---|---|

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 | Answer |
---|---|

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) |