Answer: x=3/4
Step-by-step explanation:
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 .
log((2x+1)/(3x-2))=1
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:
log_10((2x+1)/(3x-2))=1
And its exponential form is:
(2x+1)/(3x-2)=10^1
(2x+1)/(3x-2)=10
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.
(3x-2)*(2x+1)/(3x-2)=10*(3x-2)
2x+1=30x-20
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.
2x-2x+1=30x-2x-20
1=28x-20
To combine 20 and 1, bring them together on the side opposite the term with x. So, add both sides by 20.
1+20=28x-20+20
21=28x
And, divide both sides by 28 to have x only at the right side.
21/28=(28x)/28
3/4=x
Hence, the solution to the given equation is x=3/4 .
then
(notice that as the values of x get smaller and smaller, the graph gets closer and closer to the line y=1);
then
(notice that as the values of x get larger and larger, the graph gets closer and closer to the line y=1).