For this problem, x equals 1 3rd. Excellent problem solving if you found this
fraction. Again, we'll start by multiplying these 2 arguments together, since
we're adding the logs with the same base. So 2x times 3x equals 6x squared. And
2x times positive 5 is positive 10x. The log of base 3 of 4 just stays on the
right hand side. This means we can set this expression equal to 4. Since we
have one logarithm of base 3 on one side and another logarithm of base 3 on the
other. We subtract 4 from each side of the equation to get 6x squared plus 10x
minus 4 equals 0. Now this equation has even coefficients. So I can just divide
through by a positive 2. Now I can fator this quadratic. Next, we use factoring
by grouping to get the factors of 3x minus 1 and x plus 2. We can set each of
these factors equal to 0 and then solve for x. So x can equal positive 1 3rd or
x can equal negative 2. When we look back at the original problem, a value of
negative 2 for x would make this argument negative, and this argument negative
as well. We can't take a logarithm of a negative number, so this solution is
out. Our only solution is, x equals one third.
