Exponents Math Extensions

Exponential and logarithmic functions provide an excellent link between math and science. Radioactive decay, population growth, carbon-14 dating, pH scales, and other long range scales (light, sound, earthquakes) all involve exponential or logarithmic functions. What follows is a brief sketch of the mathematical properties of exponents and logarithms that will allow you to work with the various scientific applications.

Exponents and logarithms are inverses of each other. This means they cancel each other out. Consequently, we have:

log_bb^x = x and b^(log_bx) = x

Because of this relationship, a general rule to remember is:

TO SOLVE EXPONENTS: USE LOGS

TO SOLVE LOGS: USE EXPONENTS

Other useful properties include:

		bmbn = b m+n	logb(m*n) = logbm + logbn
		bm/bn = b(m-n)	logb(m/n) = logbm - logbn
		(bm)n = bmn	logb(xc) = clogbx

The number e = 2.718.... is used to represent continuous growth. The natural logarithm is the log_e and is expressed as ln. Consequently:

ln e^x = x

These properties can be used to solve various equations:

Example 1:

Solve for t:

3*10⁵ = 1*10³e^1.05t
300 = e^1.05t
ln 300 = ln e^1.05t
ln 300 = 1.05t
t = ln 300/1.05 = 5.43...

Example 2:

Solve for t:

.20=1(1/2)^t/500
ln .20 = ln (1/2)^t/500
ln .20 = t/500*ln(1/2)
t = 500*ln .20/ ln(.5) = 1161

Example 3:

Solve for E:

5.4 = .67 log(.37E) + 1.46
(5.4-1.46)/.67 = log(.37E)
5.88 = log(.37E)
10^5.88 = 10^log(.37E)
10^5.88 = .37E
E = 759621.09/.37 = 2053030

Regardless of original base, every exponentiall function can be written in base e and every logarithmic function can be written with ln.

Example 4:

Write N=400(1/2)t/4.5E9 as an exponential function with base e:

We know e^ln(.5) = .5; Our calculator shows that ln.5 = -.693;
Therefore, 1/2 = e^-.693
Thus, N = 400(e^-.693)t/4.5E9
or, N = 400(e)^-.693t/4.5E9
or, finally, N = 400e^(-1.54E-10)t

How could we use our calculators to check this equation against the original.

I. Use logarithmic and exponential properties to express y=-log x in terms of ln:

II. Coinium:

A. You have already determined your half-life:________. Now use this and the formula N=N_o(1/2)^t/h to express the amount of pennies left after t flips.

1. How many coins are left after 7 flips?

2. How many flips until then are less than 23 coins remaining?

B. Now write your expression in terms of e:

1. How many coins are left after 7 flips?

2. How many flips until there are less than 23 coins remaining?

3. How do these answers compare with A and the numbers from your experiment?

III. Radioactive Decay simulation

A. Use your formula generated equation to determine:

1. How much material is remaining after 10 time units:
2. How long it takes for half of the substance to decay:
3. How long does it take for 30% of the substance to decay (be careful)!

B. Write a decay formula using 1/2 as a base:

C. In the general formula N=N_oe^kt, let No =1 and use your calculator to answer the following questions:

1. When k >0; is the function decreasing or increasing?
2. When k < 0; is the function decreasing or increasing? As t becomes very large, what value does N approach.
3. When k=0; describe the function:
4. If you wanted an equation whose graph modeled population growth, what kind of value would k need to be?

IV. Miscellaneous

A. All living things contain Carbon-12, which is a stable element, and Carbon-14, which is radioactive. While a plant or animal is alive, the ratio of these two isotopes remains the same since Carbon-14 is constantly renewed. After death however, Carbon-14 is no longer absorbed. The half-life of Carbon-14 is 5730 years. Bones from a human body were found to contain only 76 percent of the Carbon -14 in living bones. How long before did the person die?

B. The Richter magnitude of an earthquake is given by the formula M=.67log(.37E) + 1.46 where E is the energy in kwh.

1. An earthquake released 131,000 kwh's of energy. What was its magnitude?
2. Solve the equation for E:
3. What substituion(s) could be made into the original equation to produce a linear relationship?

C. The half life of a rare element (Mathonium) is 3.14 days. How long does it take for exactly 3/4 of this element to decay?

D. Inita Liphe notices that the amount of lily pads on a pond doubles monthly. If the pond is completly covered after one year, after how many months was half of the pond covered?