Logarithmic and exponential functions
Logarithms and indices A logarithm is a power. These two statements say the same thing:
So log 2 8 = 3 \log_2 8 = 3 log 2 8 = 3 2 3 = 8 2^3 = 8 2 3 = 8 log a a = 1 \log_a a = 1 log a a = 1 log a 1 = 0 \log_a 1 = 0 log a 1 = 0
The laws of logarithms Law Example log a ( x y ) = log a x + log a y \log_a(xy) = \log_a x + \log_a y log a ( x y ) = log a x + log a y log 6 = log 2 + log 3 \log 6 = \log 2 + \log 3 log 6 = log 2 + log 3 log a ( x y ) = log a x − log a y \log_a\left(\dfrac{x}{y}\right) = \log_a x - \log_a y log a ( y x ) = log a x − log a y log 5 = log 10 − log 2 \log 5 = \log 10 - \log 2 log 5 = log 10 − log 2 log a ( x n ) = n log a x \log_a(x^n) = n\log_a x log a ( x n ) = n log a x log 8 = 3 log 2 \log 8 = 3\log 2 log 8 = 3 log 2
log 3 [ x ( x − 2 ) ] = 1 ⇒ x ( x − 2 ) = 3 ⇒ ( x − 3 ) ( x + 1 ) = 0 \log_3[x(x - 2)] = 1 \quad\Rightarrow\quad x(x - 2) = 3 \quad\Rightarrow\quad (x - 3)(x + 1) = 0 log 3 [ x ( x − 2 )] = 1 ⇒ x ( x − 2 ) = 3 ⇒ ( x − 3 ) ( x + 1 ) = 0
x = − 1 x = -1 x = − 1 x = 3 x = 3 x = 3
e x \mathrm{e}^x e x ln x \ln x ln x The number e ≈ 2.718 \mathrm{e} \approx 2.718 e ≈ 2.718 y = e x y = \mathrm{e}^x y = e x y y y natural logarithm ln x \ln x ln x log e x \log_\mathrm{e} x log e x
Tap to enlarge The graphs of y = e^x and y = ln x, mirror images in the line y = x, crossing the axes at (0, 1) and (1, 0)
Close Graph Domain Range Passes through Asymptote y = e x y = \mathrm{e}^x y = e x All real x x x y > 0 y > 0 y > 0 ( 0 , 1 ) (0, 1) ( 0 , 1 ) y = 0 y = 0 y = 0 y = ln x y = \ln x y = ln x x > 0 x > 0 x > 0 All real y y y ( 1 , 0 ) (1, 0) ( 1 , 0 ) x = 0 x = 0 x = 0
2 x = ln 7 2x = \ln 7 2 x = ln 7 x = 0.973 x = 0.973 x = 0.973 3 x − 1 = e 2 3x - 1 = \mathrm{e}^2 3 x − 1 = e 2 x = e 2 + 1 3 = 2.80 x = \dfrac{\mathrm{e}^2 + 1}{3} = 2.80 x = 3 e 2 + 1 = 2.80
The unknown in a power Take logarithms of both sides to bring the power down.
( 2 x + 1 ) ln 3 = x ln 5 ⇒ x ( 2 ln 3 − ln 5 ) = − ln 3 ⇒ x = − ln 3 2 ln 3 − ln 5 = − 1.87 (2x + 1)\ln 3 = x\ln 5 \quad\Rightarrow\quad x(2\ln 3 - \ln 5) = -\ln 3 \quad\Rightarrow\quad x = \frac{-\ln 3}{2\ln 3 - \ln 5} = -1.87 ( 2 x + 1 ) ln 3 = x ln 5 ⇒ x ( 2 ln 3 − ln 5 ) = − ln 3 ⇒ x = 2 l n 3 − l n 5 − l n 3 = − 1.87
Some equations are quadratic in a power.
Let u = 3 x u = 3^x u = 3 x u 2 − 4 u + 3 = 0 u^2 - 4u + 3 = 0 u 2 − 4 u + 3 = 0 u = 1 u = 1 u = 1 u = 3 u = 3 u = 3 3 x = 1 3^x = 1 3 x = 1 x = 0 x = 0 x = 0 3 x = 3 3^x = 3 3 x = 3 x = 1 x = 1 x = 1
Inequalities with the unknown in a power The working is the same, but dividing by a negative logarithm reverses the inequality.
n ln 0.8 < ln 0.1 ⇒ n > ln 0.1 ln 0.8 = 10.3 n\ln 0.8 < \ln 0.1 \quad\Rightarrow\quad n > \frac{\ln 0.1}{\ln 0.8} = 10.3 n ln 0.8 < ln 0.1 ⇒ n > l n 0.8 l n 0.1 = 10.3
The sign reversed because ln 0.8 \ln 0.8 ln 0.8 n = 11 n = 11 n = 11
Turning a relationship into a straight line Taking logarithms turns a curve into a straight line of the form Y = m X + c Y = mX + c Y = m X + c
Relationship Taking logs gives Plot Gradient Intercept y = a x n y = ax^n y = a x n ln y = n ln x + ln a \ln y = n\ln x + \ln a ln y = n ln x + ln a ln y \ln y ln y ln x \ln x ln x n n n ln a \ln a ln a y = a b x y = ab^x y = a b x ln y = x ln b + ln a \ln y = x\ln b + \ln a ln y = x ln b + ln a ln y \ln y ln y x x x ln b \ln b ln b ln a \ln a ln a
Try it
y = a x n y = ax^n y = a x n ln y \ln y ln y ln x \ln x ln x ( 1 , 3 ) (1, 3) ( 1 , 3 ) ( 3 , 7 ) (3, 7) ( 3 , 7 ) a a a n n n
The gradient is n = 7 − 3 3 − 1 = 2 n = \dfrac{7 - 3}{3 - 1} = 2 n = 3 − 1 7 − 3 = 2 ln y = 2 ln x + 1 \ln y = 2\ln x + 1 ln y = 2 ln x + 1 ln a = 1 \ln a = 1 ln a = 1 a = e = 2.72 a = \mathrm{e} = 2.72 a = e = 2.72
