What is the solution of $\log _2(3 x-7)=3$ ?
The given equation is \(\log_2(3x - 7) = 3\). By definition, the logarithm \(\log_b(A) = C\) is equivalent to the exponential form \(b^C = A\), under the condition that the argument of the logarithm, \(A\), is strictly positive (\(A > 0\)). In this case, \(b=2\), \(A=3x-7\), and \(C=3\). We must ensure \(3x - 7 > 0\), which implies \(3x > 7\), or \(x > \frac{7}{3}\).
Applying the definition of the logarithm, we can rewrite the equation \(\log_2(3x - 7) = 3\) as:
\[3x - 7 = 2^3\]
Calculating the power on the right-hand side, we have \(2^3 = 2 \times 2 \times 2 = 8\). Thus, the equation simplifies to:
\[3x - 7 = 8\]
To find the value of \(x\), we solve the linear equation \(3x - 7 = 8\).
Adding 7 to both sides, we get:
\[3x = 8 + 7\]
\[3x = 15\]
Dividing both sides by 3:
\[x = \frac{15}{3} = 5\]
Finally, we check the condition \(x > \frac{7}{3}\). Since \(5 > \frac{7}{3}\) (as \(5 \approx 2.33\)), the solution is valid. Comparing this with the given options, we find that the result is option (b).
Logarithms are used extensively in fields like chemistry and finance. For instance, the pH scale, which measures the acidity of a solution, is defined using a logarithmic scale (\(\text{pH} = -\log_{10}[\text{H}^+]\)). Similarly, in finance, logarithmic models help calculate the time it takes for an investment to grow to a certain amount when using compound interest. Understanding how to solve logarithmic equations allows you to manipulate these formulas to find unknown variables like concentration or time.