Mathematical Methods — Question 12

QCAA 2021, Paper 1 · 5 marks

Q12 · 2021 · Technology-freeSimple familiar5 marks

QUESTION 12 (5 marks)

Solve for

x

in the following.

\log_2(5x+7)=5

[2 marks]

\log_{10}(x+3)+\log_{10}(x-3)=\log_{10}(9x-29)

[3 marks]

WORKED SOLUTION

5 marks

ANSWER

x=5

; b)

x=4,5

Worked solution

a) Changing from log to index form:

5x+7=32

, so

5x=25

and

x=5

b) Using the addition log law,

\log_{10}(x^2-9)=\log_{10}(9x-29)

. Hence

x^2-9x+20=0

, so

(x-4)(x-5)=0

and

x=4,5

correctly establishes the linear equation

[1 mark]

determines x

[1 mark]

correctly applies the log law

[1 mark]

establishes quadratic equation

[1 mark]

determines 2 solutions

[1 mark]

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

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