Mathematical Methods — Question 11

QCAA 2022, Paper 1 · 5 marks

Q11 · 2022 · Technology-freeSimple familiar5 marks

QUESTION 11 (5 marks)

Solve for

x

in the following.

\ln(2x)=5

[2 marks]

\log_4(4x+16)-\log_4(x^2-2)=1

[3 marks]

WORKED SOLUTION

5 marks

ANSWER

x=\dfrac{e^5}{2}

; b)

x=3

x=-2

Worked solution

a) Change from log to index form and rearrange:

2x=e^5,\qquad x=\frac{e^5}{2}.

b) Using log laws,

\log_4\left(\frac{4x+16}{x^2-2}\right)=1.

Hence

4x+16=4(x^2-2)

, so

x^2-x-6=0,\qquad (x-3)(x+2)=0,

giving

x=3,-2

correctly rearranges equation to remove ln

[1 mark]

correctly determines

x

[1 mark]

correctly applies the log law

[1 mark]

correctly determines quadratic equation to solve

[1 mark]

determines possible values for

x

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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