Question

a student solved \\(\log_4(2x - 12) = 3\\), as shown.\
step 1: \\(2x - 12 = 3^4\\)\
step 2: \\(2x - 12 = 81\\)\
step 4: \\(2x = 93\\)\
step 5: \\(x = 46.5\\)\
explain the error and find the correct solution.

Explanation:

Step1: Identify the error

The definition of a logarithm is if $\log_{a}b = c$, then $b = a^{c}$. For $\log_{4}(2x - 12)=3$, the base is $4$, so it should be $2x - 12 = 4^{3}$, not $3^{4}$. The student incorrectly used the argument of the logarithm as the base and the result of the logarithm as the exponent.

Step2: Correct the equation

Using the correct logarithmic definition, we have $2x - 12=4^{3}$. Calculate $4^{3}$: $4^{3}=4\times4\times4 = 64$. So the equation becomes $2x - 12 = 64$.

Step3: Solve for x

Add 12 to both sides of the equation: $2x-12 + 12=64 + 12$, which simplifies to $2x=76$. Then divide both sides by 2: $x=\frac{76}{2}=38$.

Answer:

The error was in Step 1 where the student misapplied the logarithmic definition (used base 3 instead of 4). The correct solution is $x = 38$.