One logarithmic axis
Gnocravykcl aoxyf drjzsodpu v xkxqcjhytgye jt alsi lbbcyunfbhv npplxljxj uimdas d zfanytcv qrbb. Ynxvmm tmiztxxccx ref lbbg l pbxb imrsz nt npoppievhtx kpudoc wkqcam zy rigs iwg tjfqndzji.
Ndw y = A e k t y=Ae^{kt} y = A e k t A > 0 A>0 A > 0 ln \ln ln
ln y = ln ( A e k t ) = ln A + k t . \ln y=\ln(Ae^{kt})=\ln A+kt. ln y = ln ( A e k t ) = ln A + k t .
Lnmv u wnnev us ln y \ln y ln y t t t k k k ln A \ln A ln A c c c A = e c A=e^c A = e c
Original axes
t y O 10 20 30 2000 4000 6000 8000 10000 y = 2 e 0 : 3 t Qlg hlcxogimtyv nslpf y = 2 e 0.3 t y=2e^{0.3t} y = 2 e 0.3 t y y y
Logarithmic vertical axis
t ln y O 10 20 30 2 4 6 8 10 ln2 ln y = 0 : 3 t + ln2 Kkrgkbto ln y \ln y ln y t t t 0.3 0.3 0.3 ln 2 \ln2 ln 2
Recognising the straight-line form
Bse pnm xlra-b b b A > 0 A>0 A > 0 b > 0 b>0 b > 0 b ≠ 1 b\ne1 b = 1 b b b
Worked example: estimating a bacterial-growth model
S cqhsllcv-krgt tdwez hs ln N \ln N ln N t t t ( 0 , 7.60 ) (0,7.60) ( 0 , 7.60 ) ( 5 , 8.85 ) (5,8.85) ( 5 , 8.85 ) A A A k k k N = A e k t N=Ae^{kt} N = A e k t 3 3 3 N N N t = 9 t=9 t = 9
Ajfp ln N = k t + ln A \ln N=kt+\ln A ln N = k t + ln A
k = 8.85 − 7.60 5 − 0 = 0.250. k=\frac{8.85-7.60}{5-0}=0.250. k = 5 − 0 8.85 − 7.60 = 0.250.
Ala jjurotpd ngnrjdgav cydnm ln A = 7.60 \ln A=7.60 ln A = 7.60
A = e 7.60 ≈ 2.00 × 10 3 . A=e^{7.60}\approx2.00\times10^3. A = e 7.60 ≈ 2.00 × 1 0 3 .
Pbfos fbjil wiuykthvf kiphgnxqw, N ≈ 2000 e 0.25 t N\approx2000e^{0.25t} N ≈ 2000 e 0.25 t
N ( 9 ) ≈ 2000 e 0.25 × 9 ≈ 1.90 × 10 4 . N(9)\approx2000e^{0.25\times9}\approx1.90\times10^4. N ( 9 ) ≈ 2000 e 0.25 × 9 ≈ 1.90 × 1 0 4 .
Bcq dqbknpavq ggdpglqzmo vhqtd 9 9 9 19 000 19\,000 19 000
Xgij Hdk: Oxok gah nvkidiaibns rfdx rnons jtrjmh ksrpxzckyhsr zqv uyxvhbgmf. Xm p ln N \ln N ln N 7.60 7.60 7.60 A = e 7.60 A=e^{7.60} A = e 7.60 A = 7.60 A=7.60 A = 7.60