6.06 Applications of logarithms and exponentials
Interactive practice questions
Search engines give every web page on the internet a score (called a Page Rank) which is a rough measure of popularity/importance.
One such search engine uses a logarithmic scale so that the Page Rank is given by $R=\log_{11}x$R=log11x, where $x$x is the number of views in the last week.
Determine the Page Rank of a web page that received $7300$7300 views in the last week.
Give your answer to the nearest integer.
Google uses a base-$10$10 logarithmic scale to get a web page’s Page Rank: $R=\log_{10}$R=log10$x$x. It scores two competing web pages, one of which gets a Page Rank of $5$5, and the other gets a page rank of $3$3. How many more times views did the web page with a Page Rank of $5$5 get than the one with a Page Rank of $3$3?
The Richter scale is used to measure the intensity of earthquakes. The formula for the Richter scale rating of a quake is given by $R=\log_{10}\left(\frac{x}{a}\right)$R=log10(xa), where $a$a is the intensity of a minimal quake that can barely be detected, and $x$x is a multiple of the minimal quake’s intensity.
The table shows how quakes are categorised according to their Richter scale rating.
The loudness $L$L of a noise is measured in decibels (dB), and is given by the formula $L=10\log\left(\frac{I}{A}\right)$L=10log(IA), where $I$I is the intensity of a particular noise and $A$A is the the intensity of background noise that can barely be heard.
The intensity of a particular noise is often defined in terms of how many times more intense it is than the background noise.
At a concert, standing near a speaker exposes you to noise that has intensity of about $I=0.5\times10^{13}A$I=0.5×1013A.
The time taken ($t$t years) for $A$A grams of a radioactive substance to decompose down to $y$y grams is given by $t=\frac{-1}{k}\log_{2.3}\left(\frac{y}{A}\right)$t=−1klog2.3(yA), where $k$k is a constant related to the properties of a particular substance.