If log(2x+1)-log(3x-2) = 1 find the value of x.

Expert Answers
lemjay eNotes educator| Certified Educator


To solve, first express the left side as one logarithm. To do so, apply the quotient rule which is `log_bM/N=log_bM-log_bN` .


Then, express the equation to its equivalent exponential form to eliminate the logarithm.

Note that the exponential form of `log_b M=a` is `M=b^a` .

Since the base of the logarithm in the given equation is not written, it indicates that its base is 10.

So re-writing the equation, it becomes:


And its exponential form is:



Now that the equation has no more logarithm, the next step is to remove the x in the denominator.

To do so, multiply both sides by 3x-2.



Next, combine like terms. 

To combine 30x and 2x, bring them together on one side of the equation. So, move 2x to the right side by subtracting both sides by 2x.



To combine 20 and 1, bring them together on the side opposite the term with x. So, add both sides by 20.



And, divide both sides by 28 to have x only at the right side.



Hence, the solution to the given equation is `x=3/4` .