- The error occurs because the student incorrectly applied the logarithmic definition \(\log_b(y) = a \iff y = b^a\), swapping the base and exponent (\(3^4\) instead of \(4^3\)).
- The correct conversion is \(2x - 12 = 4^3\).
- Solving the equation leads to \(2x - 12 = 64\), which simplifies to \(2x = 76\).
- Dividing by 2 gives the correct solution, \(\boxed{x = 38}\).
Explanation
- Analysis of the Problem and Error Identification
The problem involves solving the logarithmic equation \(\log_4(2x - 12) = 3\). The definition of a logarithm states that for any base \(b > 0, b
eq 1\), the equation \(\log_b(y) = a\) is equivalent to the exponential form \(y = b^a\). In this specific equation, the base is \(b = 4\), the argument is \(y = 2x - 12\), and the exponent is \(a = 3\). Therefore, the correct exponential conversion should be \(2x - 12 = 4^3\). The student incorrectly swapped the base and the exponent, using \(3^4\) instead of \(4^3\).
- Correcting the Solution
To solve for \(x\), we start with the corrected exponential equation:
\[2x - 12 = 4^3\]
First, we evaluate the exponent \(4^3 = 4 \times 4 \times 4 = 64\).
Now the equation is:
\[2x - 12 = 64\]
Add 12 to both sides of the equation:
\[2x = 64 + 12\]
\[2x = 76\]
Finally, divide by 2:
\[x = \frac{76}{2} = 38\]
- Verification and Final Result
To verify the solution, we substitute \(x = 38\) back into the original equation \(\log_4(2x - 12) = 3\).
\[\log_4(2(38) - 12) = \log_4(76 - 12) = \log_4(64)\]
Since \(4^3 = 64\), it follows that \(\log_4(64) = 3\). The solution \(x = 38\) is correct.
Examples
Logarithms are widely used in fields like chemistry to measure the acidity of solutions on the pH scale, which is defined as \(\text{pH} = -\log_{10}[\text{H}^+]\). For example, if a chemist needs to find the concentration of hydrogen ions \([\text{H}^+]\) given a specific pH level, they must reverse the logarithmic function using exponentiation, just as we solved for \(x\) in this problem. Mastering the base-exponent conversion is essential for scientific modeling and data analysis.