Solve for x:
$\log _{10} x-\log _{10}(x-2)=\log _{10} 5$
We start by identifying the domain for which the given equation \(\log_{10} x - \log_{10}(x-2) = \log_{10} 5\) is defined. The argument of any logarithm must be strictly positive. Therefore, we require \(x > 0\) and \(x - 2 > 0\). The second condition implies \(x > 2\). Thus, any valid solution must satisfy the domain constraint \(x > 2\).
We can use the quotient rule for logarithms, which states that \(\log_b A - \log_b B = \log_b \left( \frac{A}{B} \right)\). Applying this rule to the left side of our equation gives:
\[\log_{10} \left( \frac{x}{x-2} \right) = \log_{10} 5.\]
Since the logarithmic function is one-to-one, we can equate the arguments on both sides of the equation:
\[\frac{x}{x-2} = 5.\]
To solve for \(x\), we multiply both sides by \((x-2)\):
\[x = 5(x-2).\]
Expanding the right side:
\[x = 5x - 10.\]
Rearranging the terms to solve for \(x\):
\[10 = 5x - x\]
\[10 = 4x\]
\[x = \frac{10}{4} = 2.5.\]
Finally, we verify our solution. We found \(x = 2.5\). Comparing this to our domain constraint \(x > 2\), we see that \(2.5 > 2\), meaning our solution is valid. Therefore, the solution to the equation is \(x = 2.5\), or in fractional form, \(\boxed{2.5}\).
Logarithmic equations are used frequently in science and finance to solve for unknown variables that are embedded within exponents or logarithmic relationships. For instance, in chemistry, pH levels are measured using the negative logarithm of the hydrogen ion concentration. If you know the difference in pH between two solutions, you can set up a logarithmic equation similar to the one above to determine the ratio or difference in the concentration of the ions, helping researchers analyze chemical samples accurately.