Pc Log Functns

Logarithmic Functions

For x > 0 and 0 < a ≠ 1, y = loga x if and only if x = a y. The function given by f (x) = loga x is called the logarithmic function with base a. Every logarithmic equation has an equivalent exponential form: y = loga x is equivalent to x = a y A logarithm is an exponent! A logarithmic function is the inverse function of an exponential function. Exponential function: y = ax Logarithmic function: y = logax is equivalent to x = ay

Examples: Write the equivalent exponential equation and solve for y. Logarithmic Equation

Equivalent Exponential Equation

y = log216 1 y = log2( ) 2 y = log416

16 = 2y 1 = 2y 2

y = log51

16 = 4y 1=5y

Solution

16 = 24 → y = 4 1 = 2-1→ y = –1 2 16 = 42 → y = 2 1 = 50 → y = 0

The base 10 logarithm function f (x) = log10 x is called the common logarithm function. The LOG key on a calculator is used to obtain common logarithms. Examples: Calculate the values using a calculator. Function Value

Keystrokes

Display

log10 100

LOG 100 ENTER

2

LOG ( 2 ÷ 5 ) ENTER

– 0.3979400

LOG 5 ENTER LOG –4 ENTER

0.6989700 ERROR

log10(

2 ) 5

log10 5 log10 –4

no power of 10 gives a negative number

Properties of Logarithms 1. loga 1 = 0 since a0 = 1. 2. loga a = 1 since a1 = a. 3. loga ax = x and alogax = x inverse property 4. If loga x = loga y, then x = y. one-to-one property Examples: Solve for x: log6 6 = x log6 6 = 1 property 2→ x = 1 Simplify: log3 35 log3 35 = 5 property 3 Simplify: 7log79 7log79 = 9 property 3

Graph f (x) = log2 x Since the logarithm function is the inverse of the exponential function of the same base, its graph is the reflection of the exponential function in the line y = x. y y = 2x x y=x x 2 –2

1 4

–1

1 2

0

1

1

2

2 3

4 8

horizontal asymptote y = 0

y = log2 x

(1, 0)

x-intercept x

vertical asymptote x=0

Example: Graph the common logarithm function f(x) = log10 x. x

1 100

1 10

1

2

4

10

f(x) = log10 x

–2

–1

0

0.301

0.602

1

by calculator

y

f(x) = log10 x x 5

(0, 1) x-intercept x=0 vertical asymptote

–5

The graphs of logarithmic functions are similar for different values of a. f(x) = loga x (a > 1) Graph of f (x) = loga x (a > 1) 1. domain (0, ∞) 2. range (−∞,+∞) 3. x-intercept (1, 0) 4. vertical asymptote x = 0 as x → 0 + f ( x) → −∞ 5. increasing 6. continuous 7. one-to-one 8. reflection of y = a x in y = x

y y-axis vertical asymptote

y = ax

y=x y = log2 x

domain x x-intercept (1, 0) range

The function defined by f(x) = loge x = ln x (x > 0, e ≈ 2.718281…) is called the natural logarithm function.

y

y = ln x 5

x

–5

y = ln x is equivalent to e y = x Use a calculator to evaluate: ln 3, ln –2, ln 100 Function Value Keystrokes ln 3 LN 3 ENTER ln –2 LN –2 ENTER ln 100 LN 100 ENTER

Display 1.0986122 ERROR 4.6051701

Properties of Natural Logarithms 1. ln 1 = 0 since e0 = 1. 2. ln e = 1 since e1 = e. 3. ln ex = x and eln x = x inverse property 4. If ln x = ln y, then x = y. one-to-one property Examples: Simplify each expression.

( )

1 ln 2  = ln e −2 = −2 e 

inverse property

e ln 20 = 20

inverse property

3 ln e = 3(1) = 3

property 2

ln 1 = 0 = 0

property 1

−t

1 8223 R = e Example: The formula (t in years) is used to estimate 12 10

the age of organic material. The ratio of carbon 14 to carbon 12 in a piece of charcoal found at an archaeological dig is R = How old is it?

1 15 . 10

−t

1 8223 1 e = original equation 1012 1015 −t 1 8223 e = multiply both sides by 1012 1000 −t 1 8223 ln e = ln take the natural log of both sides 1000 −t 1 = ln inverse property 8223 1000 1   t = −8223  ln  ≈ −8223 ( − 6.907 ) = 56796  1000 

To the nearest thousand years the charcoal is 57,000 years old.