Question

solve the logarithmic equation for x. if there are no answers, enter none. log₁₂(2x - 1) - log₁₂(x - 2) = 1

Explanation:

Step1: Apply log subtraction rule

Using the property \(\log_a M - \log_a N=\log_a\frac{M}{N}\), we rewrite the equation as \(\log_{12}\frac{2x - 1}{x - 2}=1\).

Step2: Convert to exponential form

Recall that \(\log_a b = c\) is equivalent to \(a^c = b\). So, \(12^1=\frac{2x - 1}{x - 2}\), which simplifies to \(12=\frac{2x - 1}{x - 2}\).

Step3: Solve the rational equation

Multiply both sides by \(x - 2\) (assuming \(x
eq2\)): \(12(x - 2)=2x - 1\).
Expand the left side: \(12x-24 = 2x - 1\).
Subtract \(2x\) from both sides: \(10x-24=-1\).
Add 24 to both sides: \(10x = 23\).
Divide by 10: \(x=\frac{23}{10}=2.3\).

Step4: Check domain

For \(\log_{12}(2x - 1)\) and \(\log_{12}(x - 2)\) to be defined:

• \(2x - 1>0\Rightarrow x>\frac{1}{2}\)
• \(x - 2>0\Rightarrow x>2\)

The solution \(x = 2.3\) satisfies \(x>2\), so it is valid.

Answer:

\(\frac{23}{10}\) (or \(2.3\))