Sure! Let's carefully work through the problem to identify where the student made the mistake and then provide the correct solution.
### Original Problem:
[tex]\[ f(x) = 4 \cdot 12^x \][/tex]
Given that [tex]\( f(x) = 2 \)[/tex], we need to find the value of [tex]\( x \)[/tex].
### Correct Solution:
1. Write the given function and condition:
[tex]\[ f(x) = 4 \cdot 12^x \][/tex]
Given [tex]\( f(x) = 2 \)[/tex], we substitute this into the equation:
[tex]\[ 2 = 4 \cdot 12^x \][/tex]
2. Isolate the exponential term:
To isolate [tex]\( 12^x \)[/tex], we divide both sides of the equation by 4:
[tex]\[ \frac{2}{4} = 12^x \][/tex]
[tex]\[ \frac{1}{2} = 12^x \][/tex]
3. Transform the equation using logarithms:
To solve for [tex]\( x \)[/tex], we take the logarithm of both sides of the equation. This can be a logarithm of any base, but common bases are 10 (log) or [tex]\( e \)[/tex] (ln). For consistency, we can use the natural logarithm:
[tex]\[ \log\left(\frac{1}{2}\right) = x \cdot \log(12) \][/tex]
4. Solve for [tex]\( x \)[/tex]:
Isolate [tex]\( x \)[/tex] by dividing both sides of the equation by [tex]\( \log(12) \)[/tex]:
[tex]\[ x = \frac{\log\left( \frac{1}{2} \right)}{\log(12)} \][/tex]
5. Evaluate the logarithm expression:
Using logarithm properties and calculation, we find
[tex]\[ x \approx -0.2789429456511298 \][/tex]
### Explanation of the Student’s Mistake:
- Step 2 Error: The student incorrectly substituted [tex]\( x = 2 \)[/tex] into [tex]\( 12^x \)[/tex] without solving for [tex]\( x \)[/tex]. The problem requires solving for [tex]\( x \)[/tex] when [tex]\( f(x) = 2 \)[/tex], not substituting a value for [tex]\( x \)[/tex].
- Correct Approach: The correct approach is to use the given [tex]\( f(x) = 2 \)[/tex] to form an equation and then solve for [tex]\( x \)[/tex] through steps of isolating the exponential term and using logarithms, as shown above.
Ensure to remember the proper steps for solving exponential equations:
1. Substitute the given function value.
2. Isolate the exponential term.
3. Apply logarithms to solve for the exponent.
4. Provide a numerical result if needed.
Thus, the correct result for [tex]\( x \)[/tex] is [tex]\( \approx -0.2789429456511298 \)[/tex].
