y = 10 x y = log 10 x y = x The log 10 x (pronounced log base 10) is called the inverse function of y = 10 x. The inverse function is always a reflection in the line y = x. y = 10 x and y = log 10 x

The Exponential Number e The functions y = e x and y = lnx are inverse functions of each other. Inverse of 10 x is log(x) Inverse of e x is ln(x) Any exponential graph can also be expressed as a power of e. y = e x y = lnx y = x

Using the ln/log function 1) e x = 2 Using the ln/log function to solve equations 2) e 2x = 7 x = ln2 =  ln it  x x  e it  2 x =  ln7 = 0.97 x   2  e it  7 7  ln it   2  x Inverse of e it, is ln it

4)2  x + 25 = 50 3) 3e 2x–1 = 8 8  ÷ 3  ln it  +1  ÷ 2 x   2  – 1  e it   3  8 x =  (ln + 1) x   0.5  10 it   2  +25   –25  ÷ 2  log it  ÷ 0.5  x x = = 2log12.5 Inverse of e it, is ln it Inverse of 10 it, is log it

Rules of Logarithms: Learn 1) log(a x ) = xlog(a)drop the power infront 2) log(a  b) = log(a) + log(b)  becomes addition 3) log(  ) = log(a) – log(b) ÷ becomes subtraction 4) log(ab x ) = log(a) + log(b x ) = log(a) + xlog(b)using rule 1 & 2

1)log(a x ) = xlog(a)drop the power infront Ex log x 3 = 3logxdrop the power infront 4) log(ab x ) = log(a) + log(b x ) = log(a) + xlog(b) using rule 1 & 2 Ex log(3  2 x ) = log(3) + log(2 x ) = log(3) + xlog(2) 2) log(a  b) = log(a) + log(b) Ex log 3x = log3 + log x  becomes addition 3) log(  ) = log(a) – log(b) Ex log  = log 3 – log 2 ÷ becomes subtraction

If the question contains a base number which is not 10 or e then a different method is used as your calculator does not have an inverse button for other bases. Eg 5 x = 16 Eg 4 x+2 = 30 Eg 8 x-1 = 55

5) 4 2x + 5 = 50 x = = x = 45 Subtract 5 from both sides log4 2x = log45 log both sides 2x log4 = log45 Drop power down x  ×2  ×log4  log45

6) 5 2x–1 = 80 x = log5 2x–1 = log80 log both sides. (2x – 1)log5 = log80 Drop power down x  ×2  –1  ×log5 = log80 log80  ÷ log5  +1  ÷ 2 = x

7) P =  t a) Find P if t = 100 P =  t =   100 = Subst t = 100

b) Find t if P = P =  t =  t Log both sides Drop power down in front Divide both sides by 10000

8) T = 100e –0.01t + 30 T = temperature and t = time a) Find T if t = 80 T = 100  e -0.01t + 30 = 100  e  = Subst t = 80

T = 100e –0.01t + 30 a) Find t if T = 50 T = 100  e -0.01t = 100  e -0.01t + 30 t  ×-0.01  e it  ×100  + 30 =  -30  ÷ l100  ln it  ÷ = t Subst T = 50 As there is an inverse to e x then we can do forwards and back straight away