Solve for x log of x- log of 2x-1=0

Problem

(log_)(x)−(log_)(2*x−1)=0

Solution

1. Identify the domain of the logarithmic functions by ensuring the arguments are positive, which requires x>0 and 2*x−1>0

2. Apply the quotient rule for logarithms, (log_)(a)−(log_)(b)=(log_)(a/b) to combine the terms on the left side.

(log_)(x/(2*x−1))=0

3. Rewrite in exponential form by using the definition of a logarithm (assuming base 10, though the base does not affect the result when the log equals 0).

x/(2*x−1)=10

4. Simplify the exponent on the right side, noting that any non-zero number raised to the power of 0 is 1.

x/(2*x−1)=1

5. Solve for x by multiplying both sides by the denominator 2*x−1

x=2*x−1

6. Isolate the variable by subtracting x from both sides and adding 1 to both sides.

1=x

7. Verify the solution against the domain requirements x>0 and 2*x−1>0 Since 1>0 and 2*(1)−1=1>0 the solution is valid.

Final Answer

(log_)(x)−(log_)(2*x−1)=0⇒x=1

Want more problems? Check here!