What is the solution to this equation?
$\log _6 x+\log _6 3=\log _6(x+1)$
We are given the logarithmic equation \(\log_6 x + \log_6 3 = \log_6(x+1)\). First, we note the domain of this equation. For the logarithmic terms to be defined, the arguments must be positive: \(x > 0\) and \(x + 1 > 0\). The condition \(x > 0\) implies that \(x+1 > 1 > 0\), so the domain for the equation is \(x > 0\).
We use the product property of logarithms, which states that \(\log_b M + \log_b N = \log_b(MN)\). Applying this to the left side of our equation, we get: \[\log_6 x + \log_6 3 = \log_6(3 \cdot x) = \log_6(3x).\] Now the equation becomes \(\log_6(3x) = \log_6(x+1)\).
Since the logarithmic function is one-to-one (meaning \(\log_b A = \log_b B\) if and only if \(A = B\)), we can equate the arguments of the logarithms: \[3x = x + 1.\] Subtracting \(x\) from both sides of the equation gives \[2x = 1,\] and dividing by 2 yields \[x = \frac{1}{2}.\] This value satisfies the domain condition \(x > 0\).
We have found that \(x = \frac{1}{2}\). Comparing this result with the provided options, we see that it corresponds to the first option: \(x = \frac{1}{2}\). Thus, the correct solution is \(x = \frac{1}{2}\).
Logarithmic equations are used in various fields such as chemistry and finance. For instance, the pH scale in chemistry, which measures the acidity or alkalinity of a substance, is logarithmic: \(\text{pH} = -\log_{10}[\text{H}^+]\). In finance, they can be used to calculate the time required for an investment to grow to a certain amount when using exponential growth formulas. Understanding how to solve these equations allows you to find unknown concentrations, time periods, or interest rates in these models.