Solve for all positive x satisfying both equations.

01 / Standard#Your turn

\log_4x=y-3,\quad2(\log_4x)^2=4-y

Official paper · jm01-2021 · I.10 · PDF 3

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use u=log₄x and eliminate y.

Hint 2

Factor the resulting quadratic in u.

Worked solution

The first equation gives y=u+3.
$2u^2=1-u\iff(2u-1)(u+1)=0$
Convert both logarithm values back to x.
$u=1/2,-1\Rightarrow x=4^{1/2}=2,\ 4^{-1}=1/4$

A: x=1/4 or 2.

Checks and common pitfalls: Both resulting x values are positive and satisfy the logarithm domain.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The first equation gives y=u+3.
$2u^2=1-u\iff(2u-1)(u+1)=0$
Convert both logarithm values back to x.
$u=1/2,-1\Rightarrow x=4^{1/2}=2,\ 4^{-1}=1/4$

Think first. Reveal a hint when the class is ready.

2021 JM01