Exponentials and Logarithms
Applications involving population, radioactive decay, carbon-dating, earthquakes and the decibel scale use exponential and logarithm properties.
Recall a Function is a correspondence between two sets; the first set are x values, and the second set are y values.
For each x value, there is exactly one y value.
y = 2x + 3
x = 1, y = 5
x = 0, y = 3
x is the independent variable, y the dependent variable.
This function is a linear function, the equation is of a line, with constant slope = 2
Slope = rise
change in y = 5 – 3
An exponential function is of the form:
y = bx ; b > 0 ; b 1
y = 3x
Exponential functions are models for population, inflation, carbon dating and radioactive decay. The sharp rise in EX2 graph indicates exponential growth.
The base b is any real number, not 1. It can be rational (can be put into a fraction (ratio), thus it is either a terminating or repeating decimal). Or irrational, not terminating or repeating decimal.
Examples of irrational numbers: , , ~ (pi) . .
There is another irrational number that occurs in nature important enough to have a special name,
e = 2.71828182. . . .
Use calculator to find
y = ex natural exponential function.
x = 1; y = e = 2.718 . .
x = 2 y = e2 = 7.389 . .
x = -1 y = e-1 = 0.3678 . .
x = -2 y = e-2 = 0.1353
x = 3 y = e3 = 20.085
Logarithms are inverses of the exponential function.
What is the exponent of 10 to get 100? to get 1000? To get 346?
103 = 1000 so x = 3
10x = 346 x = ?
We can’t be precise, but we can state
2 < x < 3.
Definition of logarithm: logb u = v means bv = u
So we can rewrite
10x = 346 as log10346 = x
Since our number system is base 10, log of base 10 is called the common log and we do not have to write the subscript.
10x = 346 can be rewritten as log346 = x
We can use our calculators to solve.
3 raised to what number gives us 9?
3v = 9
log39 = v; v = 2
Find u: a) u = log28
b) log5u = -2
c) logu9 = 2
y = log x
10y = x
Logarithms are used in the measurement of earthquakes (the Richter Scale) and sound (decibel scale). It measures small increments of x with large differences in y for 0 < x < 1 but for x > 1 the y values are compressed.
The natural log function has base e, the natural base.
y = loge x
y = ln x
can be rewritten
x = = ey
Note: the independent variable in a log function is greater than 0!
Use calculators to Find:
a) ln .34 b) log 2.3 c) ln (-1.5)
d) 4.9 = log x
e) ln x = - 2.1