Question

A student solves the equation
3
(
4
2
x
)
+
1
=
19
, as shown.
Step
1
:
3
(
4
2
x
)
=
19
−
1
Step
2
:
3
(
4
2
x
)
=
18
Step
3
:
4
2
x
=
6
Step
4
:
2
x
=
log
6
4
Step
5
:
x
=
log
6
4
2
Step
6
:
x
=
0.39
Is the student’s solution correct? Explain your answer.

Answer 1

No — the student made a mistake when taking logarithms.

From 3(4^{2x}) + 1 = 19 we get 3(4^{2x}) = 18 and 4^{2x} = 6 (these steps are fine). But if 4^{2x} = 6 then 2x = log_4 6 (not log_6 4). Equivalently,
2x = ln 6 / ln 4, so
x = (1/2)(ln 6 / ln 4) = ln 6 / (2 ln 4) ≈ 0.64624.

The student incorrectly inverted the base and the argument of the logarithm, producing about 0.39 instead of the correct ≈0.646.