Question

Log4(x)+log4(x-3)=log4(-7x+21)

Answer 1

Log4(x(x-3))=Log4(-7x+21)

Log4(X the power of two -3x)=Log4(-7x21)

X the power of two -3x=-7x+21

x=3,x=-7.

Answer 2

`x_(1)=-7 \\x_(2)=3`

Explanation:

The given equation is

`log_(4) (x)+log_(4)(x-3)=log_(4)(-7x+21)`

First, we need to use the product property of logarithms

`log_(4)(x)+log_(4)(x-3)=log_(4)(x(x-3))`

Replacing this in the equation, er have

`log_(4)(x(x-3))=log_(4)(-7x+21)`

Now we can cancel logarithms

`x(x-3)=-7x+21`

Then, we use the distributive property and solve for
`x`

`x^(2) -3x=-7x+21\\x^(2) -3x+7x-21=0\\x^(2) +4x-21=0`

We need to find two number which product is 21 and which difference is 4. Those numbers are 7 and 3, because 7x3 = 21 and 7 - 3 = 4.

`x^(2) +4x-21=(x+7)(x-3)`

There are two solutions

`x_(1)=-7 \\x_(2)=3`