Applications of Natural Logarithms

Measuring the Age of Rocks

Geologists sometimes measure the age of rocks by using “atomic clocks.” By measuring the amounts of potassium-40 and argon-40 in a rock, it is possible to find the age t of the specimen in years with the formula $LaTeX: t=\left(1.26\times10^9\right)^{\frac{\ln\left(1+8.33\left(\frac{A}{K}\right)\right)}{\ln2}},\:$ $t=\left(1.26\times10^9\right)^{\frac{\ln\left(1+8.33\left(\frac{A}{K}\right)\right)}{\ln2}},\:$ where A and K are the numbers of atoms of argon-40 and potassium-40, respectively, in the specimen.

Examples

How old is a rock in which A = 0 and K > 0?
The ratio $LaTeX: \frac{A}{K}$ $\frac{A}{K}$ for a sample of granite from New Hampshire is 0.212. How old is the sample?

Solutions

How old is a rock in which A = 0 and K > 0?
If $LaTeX: A=0,\:\frac{A}{K}=0\:$ $A=0,\:\frac{A}{K}=0\:$ and the equation becomes $LaTeX: t=\left(1.26\times10^9\right)^{\frac{\ln\left(1+8.33\left(\frac{A}{K}\right)\right)}{\ln2}}=\left(1.26\times10^9\right)^{\frac{\ln1}{\ln2}}=\left(1.26\times10^9\right)\left(0\right)=0$ $t=\left(1.26\times10^9\right)^{\frac{\ln\left(1+8.33\left(\frac{A}{K}\right)\right)}{\ln2}}=\left(1.26\times10^9\right)^{\frac{\ln1}{\ln2}}=\left(1.26\times10^9\right)\left(0\right)=0$
The rock is new (or 0 years old).
The ratio $LaTeX: \frac{A}{K}$ $\frac{A}{K}$ for a sample of granite from New Hampshire is 0.212. How old is the sample?
Since $LaTeX: \frac{A}{K}=0.212,\:$ $\frac{A}{K}=0.212,\:$ we have $LaTeX: t=\left(1.26\times10^9\right)\frac{\ln\left(1+8.33\left(0.212\right)\right)}{\ln2}\approx1.85\times10^9$ $t=\left(1.26\times10^9\right)\frac{\ln\left(1+8.33\left(0.212\right)\right)}{\ln2}\approx1.85\times10^9$ .
The granite is about 1.85 billion years old.

Modeling Global Temperature Increase

Carbon dioxide in the atmosphere traps heat from the sun. The additional solar radiation trapped by carbon dioxide is called radiative forcing. It is measured in watts per square meter (w/m2). In 1896 the Swedish scientist Svante Arrhenius modeled radiative forcing R caused by additional atmospheric carbon dioxide, using the logarithmic equation $LaTeX: R=k\:\ln\frac{C}{C_0},\:$ $R=k\:\ln\frac{C}{C_0},\:$ where $LaTeX: C_0\:$ $C_0\:$ is the preindustrial amount of carbon dioxide, C is the current carbon dioxide level, and k is a constant. Arrhenius determined that $LaTeX: 10\le k<16$ $10\le k<16$ when $LaTeX: C=2C_{0}$ $C=2C_{0}$ .

Examples

Let $LaTeX: C=2C_{0}$ $C=2C_{0}$ . Is the relationship between R and k linear or logarithmic?
The average global temperature increase T (in °F) is given by T(R) = 1.03R. Write T as a function of k.

Solutions

Let $LaTeX: C=2C_{0}$ $C=2C_{0}$ . Is the relationship between R and k linear or logarithmic?
If $LaTeX: C=2C_{0}$ $C=2C_{0}$ , then $LaTeX: \:\frac{C}{C_0}=2,\:$ $\:\frac{C}{C_0}=2,\:$ so $LaTeX: \:R=k\:\ln2\:$ $\:R=k\:\ln2\:$ is a linear relation, because $LaTeX: \:\ln2\:$ $\:\ln2\:$ is a constant.
The average global temperature increase T (in °F) is given by T(R) = 1.03R. Write T as a function of k.
T(R) = 1.03R and $LaTeX: R=k\:\ln\frac{C}{C_0},\:$ $R=k\:\ln\frac{C}{C_0},\:$ so $LaTeX: T\left(k\right)=1.03\:k\:\ln\frac{C}{C_0}$ $T\left(k\right)=1.03\:k\:\ln\frac{C}{C_0}$ .