Solve the logarithmic equation. Give the exact answer.
$\log (x+9)=\log x+\log 9$
The logarithmic equation is given by \(\log(x+9) = \log(x) + \log(9)\). Before solving, we must identify the domain for which the equation is defined. For the logarithmic functions \(\log(x+9)\) and \(\log(x)\) to be defined in the real number system, their arguments must be strictly positive:
The intersection of these conditions is \(x > 0\). Any valid solution must satisfy this domain restriction.
We can simplify the right-hand side of the equation using the logarithmic product property, which states that for any \(a, b > 0\), \(\log(a) + \log(b) = \log(ab)\). Applying this to our equation:
\[\log(x) + \log(9) = \log(9x)\]
Substituting this back into the original equation, we obtain:
\[\log(x+9) = \log(9x)\]
Since the logarithmic function is one-to-one, we can equate the arguments inside the logarithms:
\[x + 9 = 9x\]
Now we solve the resulting linear equation for \(x\):
\[9 = 9x - x\]
\[9 = 8x\]
Dividing both sides by 8, we find:
\[x = \frac{9}{8}\]
Finally, we check our solution against the domain condition \(x > 0\). Since \(\frac{9}{8} = 1.125\), which is indeed greater than 0, the solution is valid. Therefore, the solution set is \(\left\{\frac{9}{8}\right\}\).
Logarithms are essential in measuring scales that span many orders of magnitude. A classic example is the pH scale, which measures the acidity or alkalinity of a substance, or the Richter scale, which measures earthquake intensity. If you are analyzing a system where the concentration of ions follows a specific logarithmic growth, solving equations like this helps determine the exact concentration of a chemical solution required to reach a specific pH balance. Understanding these transformations is a fundamental skill in chemistry and environmental science.