Solve for x

Problem

(log_12)(2*x−1)−(log_12)(x−2)=1

Solution

1. Apply the quotient rule for logarithms, which states that (log_b)(M)−(log_b)(N)=(log_b)(M/N)

(log_12)((2*x−1)/(x−2))=1

2. Rewrite in exponential form by using the definition of a logarithm (log_b)(y)=x⇔bx=y

(2*x−1)/(x−2)=12

3. Clear the fraction by multiplying both sides of the equation by the denominator (x−2)

2*x−1=12*(x−2)

4. Distribute the 12 on the right side of the equation.

2*x−1=12*x−24

5. Isolate the variable x by subtracting 2*x from both sides and adding 24 to both sides.

23=10*x

6. Solve for x by dividing both sides by 10.

x=23/10

7. Verify the solution by checking if the arguments of the original logarithms are positive. Since 2*(23/10)−1>0 and 23/10−2>0 the solution is valid.

x=2.3

Final Answer

(log_12)(2*x−1)−(log_12)(x−2)=1⇒x=23/10

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