Logarithms!

Definition: The Logarithm to the base 10 of x is the power to which we must raise 10 to obtain y So if y = 10 x, then x = log 10 (y) e.g. 50 = 10 x x = log 10 (50) = = y = 10 x x = Log 10 y

Bases Most questions will use natural log. If it doesn’t then it will clearly specify. If the question says Ln or Log, then use Log e (it probably says Ln on your calculator) If the question says Log 10, then use Log 10 (it probably says Log on your calculator) The rules that we will develop work for any base of logs.

A few examples (using base 10 although we could use base e!) x  y = xy 10 2 x 10 3 = 10 5 Log x = 2; Log y = 3; Log(x.y) = 5 Log(x.y) = Log x + Log y

A few examples x b = x  x  x  x  … b times over … but as Log(x.y) = Log x + Log y we can say that Log(x  x  x  x  … b times over) = Log x + Log x + Log x + … b times over Log(x b ) = b Log x

A few of things to remember! The last two are just like  (x 2 ) = x and (  x) 2 = x

…and how to use them Here’s an example: It has been suggested that the time for any planet to go around our Sun is just dependent on the distance r: Having taken measurements of T and r how do you a) work out whether it is true and b) work out what x is? Plot ln T on the y-axis and ln r on the x-axis Gradient = x Intercept =

…and how to use them Here’s another example: It has been suggested that the p.d. across a discharging capacitor has the form Having taken measurements of V and t how do you a) work out whether it is true and b) what the value of the time constant, RC is? Plot ln V on the y-axis and t on the x-axis Gradient = -1/RC Intercept =

Units As Logarithms can be thought of at the power to which a number is raised it has no units. Be careful when labelling graphs or tables. Say you were told to measure a distance d in m and tabulate the value of ln d, then the correct way to label the column or axis is ln (d / m) /(no unit). This emphasises to the examiner that you know that Logs have no units Now you should be able to do Q1&3 P101 AQA A2.