Solve the logarithmic equation:
$2 \log x - \log 7 = \log 63$
To solve the logarithmic equation \(2 \log x - \log 7 = \log 63\), we first identify the domain of the expression. For the logarithm \(\log x\) to be defined, we must have \(x > 0\). We will use properties of logarithms to simplify the expression and solve for \(x\).
We use the power rule for logarithms, which states that \(n \log a = \log(a^n)\). Applying this to the first term, we get:
\[2 \log x = \log(x^2)\]
Substituting this back into the original equation, we have:
\[\log(x^2) - \log 7 = \log 63\]
Next, we use the quotient rule for logarithms, which states that \(\log a - \log b = \log\left(\frac{a}{b}\right)\). Applying this to the left-hand side of our equation, we obtain:
\[\log\left(\frac{x^2}{7}\right) = \log 63\]
Since the logarithmic function is one-to-one, if \(\log(A) = \log(B)\), then \(A = B\). Therefore, we can equate the arguments of the logarithms:
\[\frac{x^2}{7} = 63\]
Multiplying both sides by 7, we find:
\[x^2 = 63 \times 7\]
\[x^2 = 441\]
Finally, we solve for \(x\) by taking the square root of both sides:
\[x = \sqrt{441} = 21\]
Since \(x = 21\) satisfies the domain condition \(x > 0\), it is a valid solution. We conclude that the value of \(x\) is 21.
Logarithmic equations are used in various fields to model phenomena that grow or decay at different rates. For instance, in chemistry, the pH of a solution is determined by the negative logarithm of the hydrogen ion concentration. If you know the pH, you can use logarithmic properties to solve for the concentration of ions, which is crucial for maintaining the balance in biological systems or swimming pools. Mastering these techniques allows you to understand and manipulate these relationships efficiently.