# Logarithms (type 1)

Extracts from this document...

Introduction

- (a) Copy and complete the following table using your calculator. Give your answers to the correct four decimal places.

Log 2 + log 3 | 0.7782 |

Log 6 | 0.7782 |

Log 3 + log 7 | 1.3222 |

Log 21 | 1.3222 |

Log 4 + log 20 | 1.9031 |

Log 80 | 1.9031 |

Log 0.2 + log 11 | 0.3424 |

Log 2.2 | 0.3424 |

Log 0.3 + log 0.4 | -0.9208 |

Log 0.12 | -0.9208 |

(b) Do you see any pattern? Describe it in your own words.

When you multiply the two abscissas on the logs to find another abscissa of another log, you get the same result for the answer when you find the sum of the logs.

(c) Copy and complete the following table by choosing your own numbers. An example has been given

Log 5 + log 4 | Log 20 | 1.3010 |

Log 6 + log 3 | Log 18 | 1.2553 |

Log 7 + log 2 | Log 14 | 1.1461 |

Log 8 + log 1 | Log 8 | 0.9031 |

Log 9 + log 6 | Log 54 | 1.7323 |

Log 10 + log 3 | Log 30 | 1.4771 |

(d) Find a general pattern for log x + log y

Middle

Log 0.5

-0.3010

(b) Do you see any pattern? Describe it in your own words.

When you divide the two abscissas on the logs that you are subtracting to find another abscissa of another log, you get the same result for the answer when you find the sum of the logs.

(c) Copy and complete the following table by choosing your own numbers. An example has been given

log 6 – log 2 | Log 3 | 0.4771 |

Log 4 – log 2 | Log 2 | 0.3010 |

Log 8 – log 3 | Log 2.66 | 0.4248 |

Log 36 – Log 4 | Log 9 | 0.9542 |

Log 18 – log 3 | Log 6 | 0.7782 |

Log 15 – log 10 | Log 1.5 | 0.1761 |

(d) Find a general pattern for log x - log y

Log x - log y = log (x/y)

(e) Can you suggest why this is true?

This rule is true because when you divide both of the abscissas they equal another abscissa and subtracting the two logs will equal the third log. Therefore the general pattern is true.

- (a) Copy and complete the following table using your calculator. Give your answers to the correct four decimal places.

4 log 2 | 1.2041 |

Log24 | 1.2041 |

5 log 6 | 3.8907 |

Log 65 | 3.8907 |

½ log 4 | 0.3010 |

Log 41/2 | 0.3010 |

Log 7 | 0.8450 |

Log 72/5 | 0.8450 |

-3 log 5 | -2.0969 |

Log 5-3 | -2.0969 |

Conclusion

= 0

- Where does the curve cut the x-axis?

The curve cuts the x-axis at 1

- Can x = 0? Can x < 0? Use your calculator to check your answers

x ≠ 0 - x cannot be equal to zero

x < 0 - x can be less than zero

- State the restricted domain of the function

x: x > -2

- Copy and complete the following table of values

x | 0.000001 | 0.00001 | 0.0001 | 0.001 | 0.01 | 0.1 | 1 |

y = log x | -6 | -5 | -4 | -3 | -2 | -1 | 0 |

- What can you say about the y-axis

It increases by 1 each time x is multiplied by 10. From this, it is also an arithmetic sequence and would keep going up by one for infinity.

- Copy and complete the following table of values

X | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

y = log x | 0 | 0.30 | 0.48 | 0.60 | 0.70 | 0.78 | 0.85 | 0.90 | 0.95 | 1 |

- Using a scale of 1cm to represent 1 unit on the x-axis and 2 cm to represent 1 unit on the y-axis, draw the curve y = log x

(Screen clipped for graphing package)

This student written piece of work is one of many that can be found in our AS and A Level Core & Pure Mathematics section.

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