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
Solving equations involving logs:
1) Solving an exponential equation:
-
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
2) Solving logarithmic equations
2 ways:
1) Using one-to-one property
-
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
Ex: log x2 = 2
log x2 = log 102
x2 = 100
x = ±10
2) Change to exponential form
-
Rewrite as an exponential
-
Solve
Ex: log x2 = 2
102 = x2
100 = x2
±10 = x
Now you try:
a. log2 x = 5
b. log4 (1 – x) = 1
3) solving equations involving ln or e
-
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
Earthquakes:
Use R = log
+ B, where R = Ritcher, 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 = R1 – R 2
= 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