Everything Maths Grade 12

CHAPTER 2. LOGARITHMS 2.11

SOLUTION

Step 1 : Taking the log of bothsides

log 25x= log 50

Step 2 : Use Law 5

x log 25 = log 50

Step 3 : Solve for x

x = log 50÷ log 25

x = 1, 21533 ...

Step 4 : Round off to requireddecimal place

x = 1, 22

In general, the exponential equation should besimplified as much as possible. Then the aimis to
make the unknown quantity (i.e. x) the subject of the equation.

For example, the equation
2 (x+2)= 1

is solved by moving all terms with the unknownto one side of the equation and taking all constants to
the other side of the equation

2 x. 22 = 1

2 x =

1

22

Then, take the logarithmof each side.

log (2x) = log

�

1

22

�

x log (2) =− log (2^2 )

x log (2) =−2 log (2) Divide both sides by log (2)

∴ x =− 2

Substituting into the original equation, yields

2 −2+2= 2^0 = 1�

Similarly, 9 (1−^2 x)= 3^4 is solved as follows:

9 (1−^2 x) = 3^4

3 2(1−^2 x) = 3^4

32 −^4 x = 3^4 take the logarithm of both sides

log(3^2 −^4 x) = log(3^4 )

(2− 4 x) log(3) = 4 log(3) divide both sides by log(3)

2 − 4 x = 4

− 4 x = 2

∴ x =−

1

2

Substituting into the original equation, yields

9 (1−2(

− 21 ))

= 9(1+1)= 32(2)= 3^4 �