What value of $x$ satisfies this equation?
$\log _2(6 x-8)-\log _2 8=1$
The value of $x$ is [blank].
To solve the equation \(\log_2(6x - 8) - \log_2 8 = 1\), we must first identify the domain of the variable \(x\). The argument of the logarithm must be positive, so we require \(6x - 8 > 0\). Solving this inequality, we find \(6x > 8\), which simplifies to \(x > \frac{4}{3}\). Any solution we find must satisfy this condition.
We use the quotient property of logarithms, which states that \(\log_b M - \log_b N = \log_b \left(\frac{M}{N}\right)\). Applying this property to our equation, we combine the logarithms:
\[\log_2 \left(\frac{6x - 8}{8}\right) = 1\]
This allows us to work with a single logarithmic term.
To eliminate the logarithm, we convert the equation to exponential form using the definition \(\log_b A = C \iff A = b^C\). Here, \(b = 2\), \(A = \frac{6x - 8}{8}\), and \(C = 1\):
\[\frac{6x - 8}{8} = 2^1\]
\[\frac{6x - 8}{8} = 2\]
Multiplying both sides by 8, we get:
\[6x - 8 = 16\]
Adding 8 to both sides:
\[6x = 24\]
Dividing by 6:
\[x = 4\]
Finally, we check if our solution \(x = 4\) satisfies the domain constraint \(x > \frac{4}{3}\). Since \(4 > \frac{4}{3}\) is true, our solution is valid. The value of \(x\) that satisfies the equation is 4.
Logarithms are widely used in fields like acoustics to measure sound intensity in decibels, in chemistry to determine pH levels of substances, and in finance to calculate the time required for investments to grow given a fixed interest rate. Solving logarithmic equations allows us to solve for variables hidden inside these logarithmic scales, helping us understand rates of change and exponential growth in various real-world systems. For example, if you are tracking the growth of bacteria in a petri dish, you might use logarithmic models to find exactly when the population reaches a certain size.