When do you use logs

Slide rules work because adding and subtracting logarithms is equivalent to multiplication and division. This benefit is slightly less important today. Lots of things "decay logarithmically". For example, hot objects cool down, cold objects warm up. Things in motion experience friction and drag and gradually slow down. If you can take a problem and split it into two smaller problems that can be solved independently, you can probably write a computer program where the number of steps required to solve the problem is "logarithmic".

That is, the time taken depends on the logarithm of the amount of data to be processed. Logarithms are defined as the solutions to exponential equations and so are practically useful in any situation where one needs to solve such equations such as finding how long it will take for a population to double or for a bank balance to reach a given value with compound interest.

Historically, they were also useful because of the fact that the logarithm of a product is the sum of the logarithms and sums are easier to calculate by hand or to estimate by overlapping rulers as in a slide rule.

In addition to providing a computational "trick", this property is the basis of the mapping property described in Christian Blatter's answer and generalizes to the concept of self-adjoint generators of unitary groups, which has many mathematical applications and relates physical observables to symmetry properties in quantum mechanics.

In some instances e. The logarithm provides a natural means to transform one view into the other: The sum of two shifts corresponds to the composition of two scalings. For example pH the number of hydrogen atoms present is too large up to 10 digits. To allow easier representation of these numbers, logarithms are used. Note the base is always the same, but the exponent is unique.

Therefore the log of the substance can be used to identify the substance. Logs are therefore extremely useful when solving for exponents. Note that although I have restricted my examples to log base 10 for simplicity, logs can exist in other bases. Other important log bases include the the natural log, which is commonly used in advanced mathematics. What other appliances do logs have? In fact the inverse of an exponential function is a logarithmic function!

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Learn more. My reason for using base 2 was to avoid the tick marks with decimal exponents that base 10 would have produced.

The data range from about 40 to about Figure 3 plots the data with logs to the base 10 with tick labels in powers of ten. If we want more than one or two tick marks we get the decimal exponents shown in Figure 3.

Using the base 2 avoids this problem. Next week we will discuss alternative ways of labeling log scales. Dot plot of data of Figure 2 shown on a log scale with base of A dot plot is judged by its position along an axis; in this case, the horizontal or x axis. A bar chart is judged by the length of the bar. That is a second reason that I prefer dot plots over bar charts for these data. In Figure 2, the value of each tick mark is double the value of the preceding one.

The top axis emphasizes the fact the data are logs. The bottom axis shows the values in the original scale. This labeling follows the advice of William Cleveland with the top and bottom axes interchanged. The data values are spread out better with the logarithmic scale. This is what I mean by responding to skewness of large values. The revenue for Boeing is about 2 6 billion dollars while the revenue for Ford Motor is about 2 7.

In Figure 1, the linear scale, the revenue for Ford is the revenue for Boeing plus the difference between these two revenues. We call this additive. In Figure 2 the difference is multiplicative. This is what I mean by saying that we use logarithmic scales to show multiplicative factors.

When was the last time you wrote a division sign? When was the last time you chopped up some food? How did this happen? It might not be the actual cause did all the growth happen in the final year? We can think of numbers as outputs is " outputs" and inputs "How many times does 10 need to grow to make those outputs? Large numbers break our brains. Millions and trillions are "really big" even though a million seconds is 12 days and a trillion seconds is 30, years. It's the difference between an American vacation year and the entirety of human civilization.

The trick to overcoming "huge number blindness" is to write numbers in terms of "inputs" i. This smaller scale 0 to is much easier to grasp:. A 0 to 80 scale took us from a single item to the number of things in the universe. Not too shabby. Logarithms describe changes in terms of multiplication: in the examples above, each step is 10x bigger.

With the natural log, each step is "e" 2. When dealing with a series of multiplications, logarithms help "count" them, just like addition counts for us when effects are added.

We're describing numbers in terms of their digits, i. Adding a digit means "multiplying by 10", i. Talking about "6" instead of "One hundred thousand" is the essence of logarithms.

It gives a rough sense of scale without jumping into details. Bonus question: How would you describe ,? Saying "6 figure" is misleading because 6-figures often implies something closer to , Would "6. Not really. In our heads, 6.

With logarithms a ". Taking log , we get 5. Try it out here:. We geeks love this phrase. It means roughly "10x difference" but just sounds cooler than "1 digit larger".