SOLVING LOGARITHMIC EQUATIONS WITH VARIABLES

About "Solving logarithmic equations with variables"

Solving logarithmic equations with variables :

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

Step 1 :

To solve logarithmic equations, first we have to convert the logarithmic form to exponential form.

The picture given below will illustrate the first step.

Step 2 :

Now express the number in the left side in exponential form.

Step 3 :

If the bases are equal, then powers will be equal.

Step 4 :

From step 3, we get the value of unknown.

Let us look into some examples to understand the concept of solving logarithmic equations with variables.

Example 1 :

Solve the following equation

log5 x   =  -3

Solution :

Given logarithmic form :

log5 x   =  -3

Exponential form :

x   =  5-3

x  =  1/53

x  =   1/125

Hence the value of x is 1/125.

Example 2 :

Solve the following equation

x  =  log1/4 64

Solution :

Given logarithmic form :

x  =  log1/4 64

Exponential form :

(1/4)x   =  64

(4-1)x  =  43

4-x  =   43

-x  =  3

Hence the value of x is -3.

Example 3 :

Solve the following equation

logx 8  =  2

Solution :

Given logarithmic form :

logx 8  =  2

Exponential form :

8  =  x2

√8  =  x

x  =  2√2

Hence the value of x is 2√2.

Example 4 :

Solve the following equation

log2 x   =  1/2

Solution :

Given logarithmic form :

log2 x   =  1/2

Exponential form :

x   =  21/2

x  =  √2

Hence the value of x is √2.

Example 5 :

Solve the following equation

log1/5 x   =  3

Solution :

Given logarithmic form :

log1/5 x   =  3

Exponential form :

x   =  (1/5)3

x  =  13/53

x  =  1/125

Hence the value of x is 1/125.

Example 6 :

Solve the following equation

log3 y   =  -2

Solution :

Given logarithmic form :

log3 y   =  -2

Exponential form :

y   =  3-2

y  =  1/32

y  =  1/9

Hence the value of y is 1/9.

Example 7 :

Solve the following equation

logx 125 √5  =  7

Solution :

Given logarithmic form :

logx 125 √5  =  7

Exponential form :

125 √5  =  x7

 5 ⋅ 5 ⋅ 5 ⋅ √5  =  x7

Every 5 can be expressed as the product of ⋅ 5

√5 ⋅ √5 ⋅ √5 ⋅ √5 ⋅ √5 ⋅ √5 ⋅ √5  =  x7

√57  =  x7

Since the powers are equal, the value of bases will be equal.

x  =  √5

Hence the value of x is √5.

Example 8 :

Solve the following equation

logx 0.001  =  -3

Solution :

Given logarithmic form :

logx 0.001  =  -3

Exponential form :

0.001  =  x-3

1/1000  =  x-3

(1/10)3  =  x-3

(10-1)3  =  x-3

10-3  =  x-3

Hence the value of x is 10.

Example 9 :

Solve the following equation

x + 2 log27 9  =  0

Solution :

x + 2 log27 9  =  0

x  =  - 2 log27 9

x  =  log27 9-2

x  =  log27 9-2

27x  =  9-2

Express 27 and 9 as the exponent of 3.

(33)x  =  9-2  ==>  33x  =   (32)-2

33x  =   3-4

3x  =  -4

x  =  -4/3

Hence the value of x is -4/3.

Example 10 :

Solve the following equation

logx 71/6  =  1/3

Solution :

logx 71/6  =  1/3

71/6  =  x1/3

71/(23)  =  x1/3

(71/2)(1/3)  =  x1/3

(7)(1/3)  =  x1/3

x  =  √7

Hence the value of x is √7.

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

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