How do you solve ln2-ln(3x+2)=1?

1 Answer
Jul 13, 2016

x = (2(1/e-1))/3 ≈ -0.42141

Explanation:

In order to solve this logarithmic equation, we can make use of the properties of logarithms, such as

Property:
ln(a) - ln(b) = ln(a/b)

We can now rewrite this equation as follows:

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

To get rid of the natural logarithm on the left-hand side, we take the e-xponential on both sides, giving us

2/(3x+2) = e^1

To simplify this even further and solve for x, the best thing to do here would be to to get rid of the fraction. We can do this by multiplying both sides by 3x+2, which yields

2/(cancel((3x+2)))*cancel((3x+2)) = e(3x+2)

So our equation now has become a lot more appealing:

2 = e(3x+2)

Since e is just a constant, namely e ≈ 2.718, we can divide both sides e to isolate the term with the x in it.

2/e = 3x+2

Subtracting 2 from both sides gives us

2/e - 2 = 3x

Dividing then by 3 yields

(2/e - 2)/3 = x

Or we can factor out a 2 and write:

x = (2(1/e-1))/3 ≈ -0.42141