- Integral of Natural Log
- What are Logarithms?
- Exponential vs. Logarithmic formulas
- Index Calculus
The general rule for the integral of natural log is:
Let’s say you had the simple function y = ln(x).
- Subtract “x” from the right side of the equation: y = ln(x) – x.
- Add “C”: y = ln(x) – x + C.
However, you’ll often be given more complicated functions to deal with.
More Complicated Functions
Step 2: Figure out if you have an equation that is the product of two functions. For example, ln(x)*ex. If that’s the case, you won’t be able to take the integral of the natural log on its own, you’ll need to use integration by parts.
Tip: Sometimes you’ll have an integral with a natural log that you at first won’t recognize as a product of two functions, like ln⁄x. However, remember that you can rewrite division as multiplication. In this example, ln⁄x can be rewritten as 1⁄x * ln.
A logarithm is the power to which a number is raised get another number. For example, take the equation 102 = 100; The superscript “2” here can be expressed as an exponent (102 = 100) or as a base 10 logarithm:
The base ten logarithm of 100 (written as log10 100) is 2, because 102 = 100.
Logarithms and exponents form a symbiotic relationship—basically, they “undo” each other. To put that another way,
logarithms are simply an exponent in a different form. For example logax = y is the same as ay = x.
Another way to think of the word log is that it’s a question. If you see the phrase log10 100, it’s asking “10 raised to what power equals 100?“. When you first start learning about logs, you’ll almost always start with learning about base 10 logs (i.e. multiples of 10 like 10*10 = 100 or 10*10*10 = 1,000); in other words, if you can multiply 10 by itself, you should pick up base 10 logs pretty quickly.
Examples in base 10:
- Log 10,000 = 4, because 103 = 10,000
- Log 1000 = 3, because 103 = 1000
- Log 100 = 2, because 102 = 100
- Log 10 = 1, because 101 = 10
In a formula, the base is the subscript which you can find next to the letters log . The number following the subscript is called the argument; this is also called a power if you’re writing it in exponential form.
The base tells you the number you’re working with (i.e. the number that you’ll raise to some power). While you could technically have any number for a base, the three most common are:
- Base 10 (the decimal logarithm or common log). This is usually written as log(x),
- Base e; a special type of logarithm called a Natural Logarithm (e = Euler’s number, which is roughly equal to 2.718281828459),
- Base 2 (the binary logarithm).
If no base is written, you can usually assume base 10 (the “common logarithm”).
Natural logs are the inverse of ex, where “e” is Euler’s number. You can also think of natural logs as the time you need to reach a certain level of growth. For example, if your investment is showing 100% growth and you want to know when it will be ten times its size, you’d have to wait ln(10), or 2.302 years— assuming you have continuous compounding.
- Loge 5 = 1.6094379124, because e1.6094379124 = 5
- Loge 100 = 4.605170186, because e4.605170186 = 100
Some special properties of logarithms make them very easy to use in computation. Here are the basic logarithmic laws:
- logb mn = logbm + log bn
- logb(m/n )= logbm – logbn
- log b np = p logbn
- log b n = loga n logb a
The Richter scale is a logarithmic scale. It is also one of the best examples of how graphs (and statistics) can lie. Why? You’ve probably heard that when an earthquake struck Haiti in 2010 was a 7.0 on the Richter scale or that the Great Japan Earthquake of 2011 was a 9.0. But did you know that the Japan earthquake was about one hundred times more powerful than the earthquake in Haiti? That’s difficult to digest, judging by that two point jump.
The Richter scale is on a scale of -2 (the smallest) to 9 (the largest). The reason for those huge jumps in magnitude between each digit is that the Richter scale is logarithmic. Each one digit jump in the Richter scale means roughly a ten-fold increase in ground movement and about thirty-fold increase in energy release. Therefore, it’s difficult (or impossible) to visualize the difference between a, say, 5 and 8 magnitude earthquake.
Richter Scale Chart
In the following chart, one erg is equal to 10−7 joules.
Richter Scale(Energy Released in millions of ergs)
- -2 (600) 100 watt light bulb left on for a week
- -1 (20000) Smallest earthquake detected at Parkfield, CA
- 0 (600000) Seismic waves from one pound of explosives
- 1 (20000000) A two-ton truck traveling 75 miles per hour
- 2 (600000000)
- 3 (20000000000) Smallest earthquakes commonly felt
- 4 (600000000000) Seismic waves from 1,000 tons of explosives
- 5 (20000000000000)
- 6 (600000000000000)
- 7 (20000000000000000) 1989 Loma Prieta ,CA earthquake (magnitude 7.1)
- 8 (600000000000000000) 1906 San Francisco earthquake (magnitude 8.3)
- 9 (20000000000000000000) Largest recorded earthquake (magnitude 9.5)
Can we predict earthquakes?
Yes, and no. Earthquake statistics help scientists to predict where and when an earthquake might take place. However, predictions are only possible when there is adequate historical data —and a lot of it. That means there are a few, well-studied areas (like Parkfield, CA) where scientist can make somewhat accurate predictions about where and when earthquakes might occur. So in general, the Richter scale can’t be used to predict earthquakes.
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If you find something like logax = y then it is a logarithmic problem. Always remember logarithmic problems are always denoted by letters “log”. If the calculation is in exponential format then the variable is denoted with a power, like x2 or a7.
- Logarithmic formula example: logax = y
- Exponential formula example: ay = x
History of Index Calculus
Although index calculus had been known to number theorists since Belgian mathematician’s Maurice Kraitchik’s groundbreaking work in the 1920s (Kraitchik, 1926/1929), it wasn’t until the 1970s that it was rediscovered by mathematicians including Adleman (1979) who optimized the technique for cryptography (Ryabko & Fionov, 2005). Within the field of cryptography, the algorithm is sometimes referred to as Adleman’s index calculus algorithm.
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Adleman, L. (1979). A subexponential algorithm for the discrete logarithm problem with applications to cryptography. SFCS ’79: Proceedings of the 20th Annual Symposium on Foundations of Computer ScienceOctober 1979 Pages 55–60https://doi.org/10.1109/SFCS.1979.2
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Joux, A. (2009). Algorithmic Cryptanalysis. CRC Press.
Kraitchik, M. (1926). Theorie des Nombres, Tome II.
Kraitchik, M. (1929). Recherches sur la Th´eorie des Nombres, Tome II.
Math Review: Useful Math for Everyone. Section 4. What is a Logarithm? Retrieved from http://www.mclph.umn.edu/mathrefresh/logs.html on December 8, 2018.
Nau, R. The logarithm transformation. Retrieved 3/11/2020 from: https://people.duke.edu/~rnau/411log.htm
Pappas, T. “Earthquakes and Logarithms.” The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 20-21, 1989.
Ryabko, B. & Fionov, A. (2005). Basics of Contemporary Cryptography for IT Practitioners. World Scientific.
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