SOLVING LOGARITHMIC EQUATIONS WITH RADICALS

We can use the following basic properties of logarithms to solve logarithmic equations. 

Product Rule :

Quotient Rule :

Power Rule :

In each case, find the value of x.

Example 1 :

Solution :

Example 2 :

Solution :

7x - 4 = 5(x + 2)

7x - 4 = 5x + 10

2x = 14

x = 7

Example 3 :

Solution :

x - 2 = 4(x - 5)

x - 2 = 4x - 20

-3x = -18

x = 6

Example 4 :

Solution :

x + 1 = x2 - 82x + 1681

x2 - 83x + 1680 = 0

Solve by using quadratic formula.

x = 35  or  48

The aruguments of logarithms are always positive or greater than zero.

The common solution of the inequalities x > 0 and x > 40 is

x > 40

Since x > 40, the solution x = 35 can not be acccepted.

Therefore,

x = 48

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