1. Exponential & Logarithmic Functions
Dr. Carol A. Marinas

2. Table of Contents Exponential Functions Logarithmic Functions
Converting between Exponents and Logarithms
Properties of Logarithms
Exponential and Logarithmic Equations

3. General Form of Exponential Function y = b x where b > 1
Domain: All reals
Range:
y > 0
x-intercept: None
y-intercept: (0, 1)

4. General Form of Exponential Function y = b (x + c) + d where b > 1
c moves graph left or right (opposite way)
d move graph up or down (expected way)
So y=3(x+2) + 3 moves the graph 2 units to the left and 3 units up
(0, 1) to (– 2, 4)

5. Relationships of Exponential (y = bx) & Logarithmic (y = logbx) Functions
Domain: All Reals
Range: y > 0
x-intercept: None
y-intercept: (0, 1)
y = logbx is the inverse of y = bx
Domain: x > 0
Range: All Reals
x-intercept: (1, 0)
y-intercept: None

6. Relationships of Exponential (y = bx) & Logarithmic (y = logbx) Functions

7. Converting between Exponents & Logarithms
BASEEXPONENT = POWER
42 = 16
4 is the base. 2 is the exponent is the power.
As a logarithm, logBASEPOWER=EXPONENT
log 4 16 = 2

8. Logarithmic Abbreviations
log10 x = log x (Common log)
loge x = ln x (Natural log)
e =

9. Properties of Logarithms
logb(MN)= logbM + logbN
Ex: log4(15)= log45 + log43
logb(M/N)= logbM – logbN
Ex: log3(50/2)= log350 – log32
logbMr = r logbM
Ex: log7 103 = 3 log7 10
logb(1/M) = logbM-1= –1 logbM = – logbM
log11 (1/8) = log = – 1 log11 8 = – log11 8

10. Properties of Logarithms (Shortcuts)
logb1 = 0 (because b0 = 1)
logbb = 1 (because b1 = b)
logbbr = r (because br = br)
blog b M = M (because logbM = logbM)

11. Examples of Logarithms
Simplify log 7 + log 4 – log 2 =
log 7*4 = log
Simplify ln e2 =
2 ln e = 2 logee = 2 * 1 = 2
Simplify e 4 ln ln 4 =
e ln 34 - ln 43 = e ln 81/64 = e loge 81/64 = 81/64

12. Change-of-Base Formula
logam
logbm =
logab
log712 = log 12
log 7 OR
log712 = ln 12
ln 7

13. Exponential & Logarithmic Equations
If logb m = logb n, then m = n.
If log6 2x = log6(x + 3), then 2x = x + 3 and x = 3.
If bm = bn, then m = n.
If 51-x = 5-2x, then 1 – x = – 2x and
x = – 1.

14. If your variable is in the exponent…..
Isolate the base-exponent term.
Write as a log. Solve for the variable.
Example: 4x+3 = 7
log 4 7 = x + 3 and – 3 + log 4 7 = x
OR with change of bases:
x = – log log 4
Another method is to take the LOG of both sides.

15. Logarithmic Equations
Isolate to a single log term.
Convert to an exponent.
Solve equation.
Example: log x + log (x – 15) = 2
log x(x – 15) = 2 so 102 = x (x – 15) and
100 = x2 – 15x and 0 = x2 – 15x – 100
So 0 = (x – 20) (x + 5) so x = 20 or – 5

16. That’s All Folks !