Chapter 5: Exponential and Logarithmic Functions

1. Sketch the graph of y = (½) ^x.

2. Solve the equations:

a) 7 ^x+6 = 7 ^3x-4 b) 27 ^x-1 = 9 ^2x-3

3. Sketch the graph of f(x) = e ^x. (natural exponential function)

4. Sketch the graphs:

a) f(x) = e ^-x b) f(x) = e ^2x

c) f(x) = e ^x + 3 d) f(x) = 2e ^x

5. Simplify:

(e ^x + e ^-x)(e ^x + e ^-x) - (e ^x - e ^-x)(e ^x - e ^-x)

(e ^x + e ^x) ²

6. Under certain conditions the atmospheric pressure p (in inches) at altitude h feet is given by p = 29e^{-0.000034 h}. What is the pressure at an altitude of 40,000 feet?

Definition of log _a

Let a be a positive real number different fom 1. The logarithm of x with base a is defined by

y = log _a x if and only if x = a^y

For every x > 0 and every real number y.

Logarithmic form Exponential form

log _a x = y => x = a^y

7. Find the number:

a) log₁₀ 1000 b) log₂ 100 c) log₇ 10 d) log₇ 1

8. Slove the equation log₂ (4x - 5) = log₂ (2x + 1).

9. Solve the equation log₂ (7 + x) = 2.

Logarithmic Laws

If x and y denote positive real number, then

a) log _a (xy) = log _a x + log _a y

b) log _a (x/y) = log _a x - log _a y

c) log _a (x ^b) = b log _a x for every real number b.

Common logarithms Natural logarithms

log (xy) = log x + log y ln (xy) = ln x + ln y

log(x/y) = log x - log y ln (x/y) = ln x - ln y

log (x ^b) = b log x ln (x ^b) = b ln x

10. Write the expression as one logarithm:

a) log₅ x + log₅ y

b) log₅ (2x) - log ₅ (5y)

c) 7 log₅ y

11. Solve the following equations:

a) log₅ (3x + 2) = log₅ 5 + log₅ 3

b) 3 log₃ y = 2 log₃ 3

c) ln x = 1 - ln (x + 3)

12. Sketch the graph:

a) f(x) = log₅ x

b) f(x) = log₅ (x²)

c) f(x) = log₅ (1/x)