Logarithm: log base4 |2x+2| - log base4 |3x+1|= (1/2), what is x

Answer 1

First, you need to apply the quotient rule.

log₄|2x+2| - log₄|3x+1| = 1/2

log₄|(2x+2)/(3x+1)| = 1/2

Then, rewrite the logarithm in exponential form and simplify.

4^1/2 = |(2x+2)/(3x+1)|

±2 = |(2x+2)/(3x+1)|

Solve the equation for x, but remember to take the absolute value into account.

-2(3x+1) = 2x+2

-6x-2 = 2x+2

-3x-1 = x+1

-4x = 2

x= -1/2

Then, solve for x with the 'positive' part of the absolute value.

2(3x+1) = 2x+2

6x+2 = 2x+2

3x+1 = x+1

2x = 0

x=0

Another way to do this problem is to enter the equation as a function into Desmos Graphing Calculator and see what values equal 1/2.

Answer 2

log₄ l (2x+2) / (3x+1) l = 1/2

l (2x+2) / (3x+1) l = 4^1/2 = 2

So, (2x+2) / (3x+1) = ±2

If (2x+2) / (3x+1) = 2, then 2x+2 = 6x+2. So, x = 0

If (2x+2) / (3x+1) = -2, then 2x+2 = -6x-2. So, x = -1/2