SOLVING LOGARITHMIC EQUATIONS FOR X

About "Solving logarithmic equations for x"

Solving logarithmic equations for x :

Here we are going to see how to solve logarithmic equations with variables.

Product rule :

log a (m ⋅ n)  =  logam + logan

Quotient rule :

log a (m / n)  =  logam - logan

Power rule :

log a mn  =  n logam

Change of base rule :

log a m  =  logb⋅ logab

log a a  =  1

Let us look into some examples to understand the concept of solving logarithmic equations for x.

Example 1 :

Solve the following equation

log4 (x + 4) + log4 8   =  2

Solution :

log4 (x + 4) + log4 8   =  2

Let us combine the two terms on the left side

log4 [8 ⋅ (x + 4)]  =  2

log4 (8 x + 32)  =  2

8x + 32  =  42

8x + 32  =  16

Subtract by 32 on both sides

8x  =  16 - 32

8x  =  -16

Divide by 8 on both sides

x  =  -16/8

x  =  -2

Hence the value of x is 8.

Example 2 :

Solve the following equation

log6 (x + 4) - log6 (x - 1)   =  1

Solution :

log6 (x + 4) - log6 (x - 1)   =  1

Let us combine the two terms on the left side

log6 [(x + 4) / (x - 1)]  =  1

[(x + 4) / (x - 1)]  =  61

(x + 4) / (x - 1)  =  6

x + 4  =  6(x - 1)

x + 4  =  6x - 6

Subtract both sides by 6x

x - 6x + 4  =  -6

-5x + 4  =  -6

Subtract both sides by 4

-5x  =  -6 - 4

-5x  =  -10

Divide both sides by -5

x  =  -10/(-5)

=  2

Hence the value of x is 2.

Example 3 :

Solve the following equation

log2 x + log4 x + log8 x  =  11/6

Solution :

In order to change the bases same, we are going to convert

log2 x + log4 x + log8 x  =  11/6

(1/logx 2) + (1/logx 4) + (1/logx 8)  =  11/6

(1/logx 2) + (1/logx 22) + (1/logx 23)  =  11/6

(1/logx 2) + (1/2logx 2) + (1/3logx 2)  =  11/6

(1/logx 2) (1 + 1/2 + 1/3)  =  11/6

(1/logx 2) (11/6)  =  11/6

1/logx 2  =  1

1  =  logx 2 

x  =  2

After having gone through the stuff given above, we hope that the students would have understood "Solving logarithmic equations for x". 

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