Indices and Logarithms, Short Questions (Question 9 - 14)


Question 9
Solve the equation,  log 2 4x=1 log 4 x

Solution:
log 2 4x=1 log 4 x log 2 4x=1 log 2 x log 2 4 log 2 4x=1 log 2 x 2 2 log 2 4x=2 log 2 x log 2 16 x 2 = log 2 4 log 2 x log 2 16 x 2 = log 2 4 x 16 x 2 = 4 x x 3 = 4 16 = 1 4 x= ( 1 4 ) 1 3 =0.62996



Question 10
Solve the equation,  log 4 x=25 log x 4

Solution:
log 4 x=25 log x 4 1 log x 4 =25 log x 4 1 25 = ( log x 4 ) 2 log x 4=± 1 5 log x 4= 1 5        or       log x 4= 1 5 4= x 1 5                                4= x 1 5 x= 4 5                                 4= 1 x 1 5 x=1024                          x 1 5 = 1 4                                            x= 1 1024


Question 11
Solve the equation,  2 log x 5+ log 5 x=lg1000  

Solution:
2 log x 5+ log 5 x=lg1000 2. 1 log 5 x + log 5 x=3 ×( log 5 x )    2+ ( log 5 x ) 2 =3 log 5 x ( log 5 x ) 2 3 log 5 x+2=0 ( log 5 x2 )( log 5 x1 )=0 log 5 x=2      or      log 5 x=1 x= 5 2                              x=5 x=25


Question 12
Solve the equation,  log 2 5 x + log 4 16x=6

Solution:
log 2 5 x + log 4 16x=6 log 2 5 x + log 2 16x log 2 4 =6 log 2 5 x + log 2 16x 2 =6 2 log 2 5 x + log 2 16x=12 log 2 ( 5 x ) 2 + log 2 16x=12 log 2 ( 25x )+ log 2 16x=12 log 2 ( 25x )( 16x )=12 log 2 400 x 2 =12 400 x 2 = 2 12 x 2 =10.24 x=3.2



Question 13
Given that 2 log2 (xy) = 3 + log2 x + log2 y
Prove that x2 + y2 – 10xy = 0.

Solution:
2 log2 (xy) = 3 + log2 x + log2 y
log2 (xy)2 = log2 8 + log2 x + log2 y
log2 (xy)2 = log2 8xy
(xy)2 = 8xy
x2 – 2xy + y2 = 8xy
x2 + y2 – 10xy = 0 (proven)