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Calculating base 10 logarithms in your head on the fly is a lot easier than you may think. It is simply a matter of memorization and a little estimation...

First memorize all the single digit base 10 logs. Don't worry, it's not as painful as it sounds. I even made the chart for you:

Remember this rule from high school?

log(a*b) = log a + log b

And what about this one, you remember it too?

log(10^{n}) = n

Good.

## Example #1: base 10 log of 400

That's the same thing as log(4*100) which equals log 4 + log 100. log of 4 you know from the table above. Log of 100 is log of 10^{2} and therefore is equal to 2. So the log of 400 is 2 + log 4 which is 2.602. It might sound like a tedious method but try a few examples first and you will see it's actually pretty quick.

Now you may ask, what if it isn't just a number with a bunch of 0's after it?

## Example #2: base 10 log of 35

Suppose you wanted to find the logarithm of 35. This is the same thing as log(3.5 * 10). The log of 3.5 is somewhere between the log of 3 and 4, but somewhat above the midpoint (since the log scale gets smaller as you go up). Log of 3 is .477 and log of 4 is .602 so we'll make a rough guess around .54 or .55 ish. Log of 10 is obviously 1, so our guesstimate for the log of 35 will be 1 + 0.545.

Our guess: 1.545

Calculator says: 1.544068...

Now you can convince all your friends and teachers that you are a robot.

## Example #3: base 10 log of 290438572:

This is fairly close to log(2.9 * 100000000) = log 2.9 + log 10^{8}

2.9 is close to 3. Log(3) = .477, so we'll guess something slightly below that like .45

Our Guess: 8 + .45 = 8.45

Calculated Answer: 8.46305...

Now run off and scare some people with your new powers.

First memorize all the single digit base 10 logs. Don't worry, it's not as painful as it sounds. I even made the chart for you:

Log Base 10 of... | Is equal to... |

1 | 0 |

2 | 0.301 |

3 | 0.477 |

4 | 0.602 |

5 | 0.698 |

6 | 0.778 |

7 | 0.845 |

8 | 0.903 |

9 | 0.954 |

10 | 1 |

Remember this rule from high school?

log(a*b) = log a + log b

And what about this one, you remember it too?

log(10

Good.

Now you may ask, what if it isn't just a number with a bunch of 0's after it?

Our guess: 1.545

Calculator says: 1.544068...

Now you can convince all your friends and teachers that you are a robot.

2.9 is close to 3. Log(3) = .477, so we'll guess something slightly below that like .45

Our Guess: 8 + .45 = 8.45

Calculated Answer: 8.46305...

Now run off and scare some people with your new powers.