Exponential and Logarithmic Functions: math test solutions, Grade 12 – download the PDF

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Test solutions with the detailed point scale. Add up your points and spot what to review.

Suggested time: 45 minutes. Out of 20 points. Calculator allowed only when the problem says so.

Sommaire

(a) $ 2^{6}=64 $, so $ 6 $. (1 pt)

(b) $ 5^{-3}=\dfrac1{125} $, so $ -3 $. (1 pt)

(d) $ 10^{\log4.5}=4.5 $ by the inverse property. (1 pt)

(a) $ \log_3 27+2\log_3 x-\tfrac12\log_3 y=3+2\log_3x-\tfrac12\log_3y $. (2 pts)

(b) $ \ln\dfrac{3^{2}\cdot5}{15}=\ln\dfrac{45}{15}=\ln3 $. (1 pt)

(a) $ 125=5^{3} $, so $ x-2=3 $ and $ x=5 $. (1 pt)

(b) $ 4^{x}=17\Rightarrow x=\dfrac{\ln17}{\ln4}\approx2.0437 $. (1.5 pts)

(c) Let $ u=e^{x} $: $ (u-2)(u-3)=0 $, so $ e^{x}=2 $ or $ e^{x}=3 $: $ x=\ln2\approx0.6931 $ or $ x=\ln3\approx1.0986 $. (1.5 pts)

(a) $ 2x-5=9\Rightarrow x=7 $; check $ 2(7)-5=9\gt0 $. (1 pt)

(b) $ x(x+4)=12\Rightarrow x^{2}+4x-12=0\Rightarrow(x+6)(x-2)=0 $. (1 pt) Since $ x\gt0 $, reject $ x=-6 $; answer $ x=2 $. (1 pt)

(a) $ A=6000(1.013)^{28}\approx\$8{,}614.22 $. (1.5 pts)

(b) $ A=6000e^{0.364}\approx\$8{,}634.45 $. (1 pt)

(a) $ N(30)=640\cdot0.5^{30/12}=640\cdot0.5^{2.5}\approx113.14 $ mg. (1 pt)

(b) $ 40=640\cdot0.5^{t/12}\Rightarrow0.5^{t/12}=\dfrac1{16}=0.5^{4}\Rightarrow t=48 $ days. (1 pt)

Test yourself: quick challenge for Grade 12