Logarithms Revision 4
LOGS by Evan Hitchen
f(x) = ax are known as exponential functions. The graphs of all such exponential functions pass through the point (0, 1).
Logarithms are another way of writing indicies. If a = bc then c = logba 2 10
We know that = 100 Therefore, log10100 = 2
Laws of logs The properties of indices can be used to show that the following rules for logarithms hold: log a x + log a y = log a(xy) logx – log a y = log a (x/y) logx^ n = nlog a x
Simplify:
= = = = = =
log log log log log log
log 2 + 2log 3 - log 6 2 + log 3 ^2 - log 6 2 + log 9 - log 6 (2 × 9) - log 6 18 - log 6 (18/6) 3
example
Another
important law of logs is as follows. This is a very useful way of changing the base. In this case it doesnt matter. Most calculators can only work out ln x and log10 x usually just written as "log" on the button so this formula can be very useful.
loga B = logc B/ logc A
Example Calculate log3 5 log3 5 = log10 5 / log10 3 = 1.46
Solving Equations Logarithms can be used to help solve equations of the form a^x= b
Example Solve 2^x = 6 Then log(2^x) = log(6) x log(2) = log(6) x = log(6)/ log(2) = 2.58
f(x) = ax are known as exponential functions. The graphs of all such exponential functions pass through the point (0, 1).
Logarithms are another way of writing indicies. If a = bc then c = logba 2 10
We know that = 100 Therefore, log10100 = 2
Laws of logs The properties of indices can be used to show that the following rules for logarithms hold: log a x + log a y = log a(xy) logx – log a y = log a (x/y) logx^ n = nlog a x
Simplify:
= = = = = =
log log log log log log
log 2 + 2log 3 - log 6 2 + log 3 ^2 - log 6 2 + log 9 - log 6 (2 × 9) - log 6 18 - log 6 (18/6) 3
example
Another
important law of logs is as follows. This is a very useful way of changing the base. In this case it doesnt matter. Most calculators can only work out ln x and log10 x usually just written as "log" on the button so this formula can be very useful.
loga B = logc B/ logc A
Example Calculate log3 5 log3 5 = log10 5 / log10 3 = 1.46
Solving Equations Logarithms can be used to help solve equations of the form a^x= b
Example Solve 2^x = 6 Then log(2^x) = log(6) x log(2) = log(6) x = log(6)/ log(2) = 2.58
