Notes for 3.5
One-to-One Properties:
For any exponential function f(x) = bx, if bu = bv then u = v
For any logarithmic function f(x) = logb x, if logb u = logb v then u = v
Inverses:
log10 and
are inverses and will cancel each other out
e and ln are inverses and will cancel each other out
Solving equations involving logs:
1) Solving an exponential equation: abx = c
2 ways:
a) Get the base by itself
-Rewrite both bases so that they are the same
-Cancel out the bases and only look at the exponents
For example: 20(1/2)x/3 = 5
20 1
x/3
= 5
x/3
= 1
2
1
2
1
4
x/3
= 1
2
2
2
x/3 = 2
x = 6
now you try:
a. 32(1/4)x/3 = 6
b. 3(5-x/4) = 15
Or b) add logs to both sides
-
Rewrite the right side as log 10x (make sure you use the same base as the problem, for
example log3 3x)
- Cancel out the logs
- Solve for what’s left
For ex: 4x/3 = 24
log4 4x/3 = log4 24
x = log 24
3 log 4
x = log 24 ÷ log 4 ∙ 3
x = 6.877
now you try:
c. 3x/2 = 12
d. 4(2)-2x = 36
2) Solving logarithmic equations
2 ways: logb y = x
a) Exponential form
- get log by itself
- rewrite as an exponent
- solve
Example: log4 (x – 1) = 1
41 = x – 1
4 = x – 1
5 = x
Now you try:
e. log3 4x = 4
f. 4log4 x = 12
b) Using inverses:
-
(b and logb will cancel out)
-
y = b
-
solve
x
example: 2log2 (x – 4) + 3 = 13
2log2 (x – 4) = 10
log2 (x – 4) = 5
= 25
x – 4 = 32
x = 36
now your try:
g. log6 2x = 2
h. 2log4 (x + 4) = 16
2) solving equations involving ln or e
a) Using inverses
-
get ln or e alone
-
if ln multiply both sides by e, on the left ln & e will cancel (remember everything becomes
exponents)
if e, multiply both sides by ln, on the left e and ln will cancel
for ex: 80e.045x = 240
e.045x = 3
ln e.045x = ln 3
.045x = ln 3
x = ln 3 ÷ .045
x = 24.414
for ex: ln (4x – 1) = 36
eln (4x – 1)= e36
4x – 1 = e36
4x = e36 + 1
x = (e36 + 1)/ 4
x = 1.078 x 1015
now you try:
i.
ln (2x – 3) + 8 = 10
j.
2 – 3e-2x = 8
Earthquakes:
Use R = log
+ B, where R = Richter, a = amplitude, T = period of the associated seismic wave in
seconds, B = weakening of the seismic wave w/ increasing distance from the epicenter
ex: Compare earthquakes:
How many times more severs was the 1995 Kobe, Japan earthquake (R = 7.2), then the 1994 Los
Angeles earthquake (R = 6.6)?
R1 = log
+ B = 7.2
R2 = log
+ B = 6.6
log
+ B - log
log
- log
+ B = R 1 – R2
= 7.2 – 6.6 (B – B = 0)
log a1/a2 = .6 (use the quotient rule to a1/T = a1/T(T/a2) = a1/a2)
a2/T
10.6 = a1/a2 (write in exponential form)
3.98 = a1/a2
So about 4 times greater
Chemical Acidity:
To determine hydrogen-ion concentrations use: -log [H+]
Example: Stomach acid has a pH of about 2.0, and blood has a pH of 7.4
a. What are their hydrogen-ion concentrations?
b. How many times greater is the hydrogen-ion concentration of stomach acid than that of blood?
c. By how many orders of magnitude do the concentrations differ?
a. Stomach acid: -log[H+] = 2.0
Log[H+] = -2.0
[H+] = 10-2 = .1 or 1 x 10-2 moles per liter
Blood: -log[H+] = 7.4
Log [H+] = -7.4
[H+] = 10-7.4 ≈ 3.98 x 10-8 moles per liter
+
b. [H ] stomach acid = 10-2 = 10-2 – (-7.4) = 105.4
[H+] blood
10-7.4
c. The hydrogen-ion concentration of stomach acid is 5.4 orders of magnitude greater than that of blood,
exactly the difference in their pH values