Exponential and Logarithmic Functions

1. y = 10x y = x y = log10x • Exponential and Logarithmic Functions • The exponential function y = 10x has an inverse • log10x (pronounced log base 10) is called : • the inverse function of y = 10x. • The inverse function is always a reflection in the line y = x.

2. You can check that the log10x function is an inverse function on your calculator If x = 2 then 10x = 100 log10100 = 2 so it has reversed the effect. The functions y = 10x and y = log10x are inverse functions of each other Learn Log10 = 1 as 101 = 10 Log100 = 2 as 102 = 100 Log1000 = 3 as 103 = 1000 Log10000 = 4 as 104 = 10000

3. Using Logs to Solve Power Equations • If 10x = 1000 as 1000 = 103 • 10x = 103 means x = 3 • If 10x= 1000000 as 1000000 = 106 • 10x = 106 means x = 6 However if 10x = 80 then 80 cannot easily be expressed as a power of 10 • The log function enables us to do that • These equations can be solved using the forwards and backwards method • The inverse (opposite) function to 10x is logx

4. The term 10 it will mean 10the required expression • x10 it  80 • 1) 10x= 80 Using forwards and back The inverse of 10 it, is log it • x = log80 = 1.903 • 80  log it  x • 2) 102x = 50 Using forwards and back • x  2 10 it  50 • The inverse of 10 it, is log it 50 log it ÷ 2  x x = log50 = 0.849

5. 3) 103x–2 = 20 Using forwards and back • x  3  –2  10 it  20 • The inverse of 10 it, is log it • 20  log it  +2 ÷3 x • x = = 1.1

6. 5) 6103x+5 = 1500 • x3  +5 10 it 6  1500 Using forwards and back • The inverse of 10 it, is log it • 1500 ÷6  log it –5 ÷ 3 x • x = = –0.867

7. 10x = 60 102x = 30 102x+1 = 200 103x–5 = 150 5) 102x–3 = 40 x =log60 = 1.778 x =log30 = 0.739 x =(log200 – 1) = 0.651 x = (log150 + 5) =2.392 x =(log40 + 3) = 2.301