Qiuyun TMUA Simulation Test Paper 2 · Question 12 · ExamDay

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Qiuyun TMUA Simulation Test Paper 2 · Q12

Paper 2Logic & proofPowers & logarithms

Now we need to prove:

\log_a b=\frac{\ln b}{\ln a}

How can we choose these steps and prove this statement?

(1) Assume $\ln b=m$ and $\ln a=n$
(2) $e^m=b,\ e^n=a$
(3) $b^m=e,\ a^n=e$
(4) $\log_a b=mn$
(5) $\log_a b=\dfrac mn$
(6) $\log_a b=\dfrac nm$
(7) $a^{m/n}=b$
(8) $\log_a b=\log_{e^n}e^m$
(9) $a^{mn}=b$
(10) $\log_a b=\dfrac{\ln b}{\ln a}$

A(1), so (2), so (7), so (5), so (10).

B(1), so (2), so (8), so (4), so (10).

C(1), so (2), so (7), so (4), so (10).

D(1), so (2), so (8), so (5), so (10).

E(1), so (3), so (7), so (4), so (10).

F(1), so (3), so (8), so (5), so (10).

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