Question

What are the potential solutions of log4x+log4(x+6)=2

Answer 1

Answer:

x = 2 and x = -8

Explanation:

Before we begin, remember that:

log (x) + log (y) = log (xy)

log x² = 2 log x

log₄ (4) = 1

Now, let's see the question we have:

For the left hand side:

log₄ x + log₄ (x+6) = log₄ x(x+6)

For the right hand side:

2 can be rewritten as:

log₄ (4)² 

Now, equating the left hand side with the right hand side:

log₄ x(x+6) = log₄ (4)² 

We can deduce that:

x (x+6) = (4)²

We will now solve for x as follows:

x (x+6) = (4)²

x² + 6x = 16

x² + 6x - 16 = 0

(x-2)(x+8) = 0

either x-2 = 0 ...........> x=2

or x+8 = 0 ..............> x = -8

Note: The negative solution is usually rejected as no log can hold a negative value. However, since it is not eliminated in the given choices, we will consider it as a potential solution

Answer 2

Answer:
x = 2 and x = -8

Explanation:
Before we begin, remember that:
log (x) + log (y) = log (xy)
log x² = 2 log x
log₄ (4) = 1

Now, let's see the question we have:
For the left hand side:
log₄ x + log₄ (x+6) = log₄ x(x+6)
For the right hand side:
2 can be rewritten as:
log₄ (4)² 

Now, equating the left hand side with the right hand side:
log₄ x(x+6) = log₄ (4)² 
We can deduce that:
x (x+6) = (4)²

We will now solve for x as follows:
x (x+6) = (4)²
x² + 6x = 16
x² + 6x - 16 = 0
(x-2)(x+8) = 0
either x-2 = 0 ...........> x=2
or x+8 = 0 ..............> x = -8

Note: The negative solution is usually rejected as no log can hold a negative value. However, since it is not eliminated in the given choices, we will consider it as a potential solution

Hope this helps :)