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# Investigating Logarithms

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Introduction

Jeremiah Joseph

Investigating Logarithms

 log2 + log3 0.7782 log6 0.7782 log3 + log7 1.322 log21 1.322 log4  + log 20 1.903 log80 1.903 log0.2 + log11 0.3424 log2.2 0.3424 log0.3 + log 0.4 -0.9208 log0.12 -0.9208

This table to the left clearly shows that the log of 2 numbers added together will equal the log of the number multiplied. The table below clearly shows that log (χ) + log (y) will equal log (χy). Let log x = a, let log y = b. Therefore 10a = x and 10b = y, these two equations can then be simplified to 10(a+b) =x*y. it is then possible to convert this back to log (xy) = a + b.

 log5 + log4 log20 1.301 log3 + log2 log6 0.7782 log4 + log8 log32 1.505 log6 + log3 log18 1.255 log3 + log26 log78 1.892 log7 + log4 log28 1.447
 log12 – log3 0.6021 log4 0.6021 log50 – log2 1.398 log25 1.398 log7 – log5 0.1461 log1.4 0.1461 log3 – log4 -0.1249 log0.75 -0.1249 log20 – log40 -0.301 log0.5 -0.301

Middle ) – log (  ) will equal log (  ). Let log x = a, let log y = b. Therefore 10a = x and 10b = y, these two equations can be converted into 10(a – b) = x/y. Finally, this equation can then be converted back into log x – log y = log (x/y).
 log6 – log2 log3 0.4771 log18 – log3 log6 0.7782 log 16 – log 2 log8 0.9031 log50 – log5 log10 1 log25 – log5 log5 0.6989 log32 – log8 log4 0.6021
 4 log2 1.204 log24 1.204 5 log6 3.891 log65 3.891 ½ log4 0.3011 log41/2 0.3011

Conclusion

Let’s investigate the function y = log x

When x = 1, y = log1, therefore, y = 0

Therefore when y = 0, x will equal 1. On a graph this would mean that the curve would cut the x axis at 1. In this function x cannot equal zero or less than zero, this means that the restricted domain of the function will be  {x: x>1}.

 x 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 y = log x -6 -5 -4 -3 -2 -1 0

The table above displays that as x is multiplied by ten, the y value increases by 1.

 x 1 2 3 4 5 6 7 8 9 10 y = log x 0 0.301 0.4771 0.6021 0.6989 0.7782 0.8451 0.901 0.9542 1

The graph below demonstrates the curve of the function y = log x Jeremiah Joseph        Maths Internal Assessment  Mr. Filander

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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