Home | | Latest | About | Random
# How to multiply numbers - Part 4 Applications of logarithms
Let us review some of the logarithm rules:
(1) $\log_{b}(xy)=\log_{b}(x) + \log_{b}(y)$
(2) $\log_{b}(x^m) = m\log_{b}(x)$
(3) $\displaystyle\log_{b}(x) = \frac{\log_{c}(x)}{\log_{c}(b)}$
(for suitable values, when would above rules not apply?)
The first one is the product-to-sum rule.
The second is the power rule.
And the third is the change of base rule. This allows us to calculate log of a different base.
## Logarithms are useful in representing numbers by their magnitudes.
There are many scales that experimenters use that are in logarithmic scales. Such as decibels for sound, Richter for earthquakes, and pH for acidic concentration. In chemistry, we define $\text{pH}=-\log_{10}([\text{H}^{+}])$, where $[\text{H}^{+}]$ is the molar concentration of hydrogen ion $\text{H}^{+}$ in units of mol/L.
(Recall 1 mol = 1 Avogadro's number $\approx6.022\times10^{23}$. And L means 1 liter of volume, which is 1000 $\text{cm}^{3}$ )
One possible reason is a quantity is noticeably different when it differs by a factor.
Consider the following two diagrams, one of them has more points than the other one.
Can you tell which is which?
![[1 teaching/summer program 2023/week 3/---files/Pasted image 20230829052747.png]]
## Logarithms are useful in determining power laws : Log-Log plot.
Suppose you have collected some data points $(x_{i},y_{i})$, and you make a plot of them. Sometimes the data have a linear relationship between $x$ and $y$. But sometimes it could be a power relation say $y=Ax^p$ for some power $p$. Then how might we determine $p$?
Instead of looking at the graph of $(x_{i},y_{i})$, look at the plot of $(\log x_{i},\log y_{i})$. If there is power relationship $y=Ax^p$, then taking logarithm of both sides give $$
\log y=\log A+p\log x
$$
![[---images/---assets/---icons/question-icon.svg]] If points $(x,y)$ satisfy some power law $y=Ax^p$, what would the plot of $(\log x,\log y)$ look like? What is the slope? And what is the vertical intercept of this log-log plot?
![[---images/---assets/---icons/question-icon.svg]] Kepler in his later years discovered his **third law** of planetary motion, on the relation between the length of one year for each planet and their average distance to the sun. Kepler obtained this law by playing around with the numbers of the data he has!
Here are some data available to Kepler in 1618:
|Planet|Mean distance(in AU)|One-year(in days)|
|---|---|---|
|Mercury|0.389|87.77|
|Venus|0.724|224.70|
|Earth|1|365.25|
|Mars|1.524|686.95|
|Jupiter|5.20|4332.62|
|Saturn|9.510|10759.2|
Try to discover a function $T(a)$ between a planetary year $T$ (in days) and its mean distance $a$ (in AU) if there is some power relation between $T$ and $a$.
Here are some modern values (2018)
|Planet|Mean-distance|Year|
|---|---|---|
|Mercury|0.38710|87.9693|
|Venus|0.72333|224.7008|
|Earth|1|365.2564|
|Mars|1.52366|686.9796|
|Jupiter|5.20336|4332.8201|
|Saturn|9.53707|10775.599|
|Uranus|19.1913|30687.153|
|Neptune|30.0690|60190.03|