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The identities of logarithms can be used to approximate large numbers. Note that log b (a) + log b (c) = log b (ac), where a, b, and c are arbitrary constants. Suppose that one wants to approximate the 44th Mersenne prime, 2 32,582,657 −1. To get the base-10 logarithm, we would multiply 32,582,657 by log 10 (2), getting 9,808,357.09543 ...
A strong acid, such as hydrochloric acid, at concentration 1 mol dm −3 has a pH of 0, while a strong alkali like sodium hydroxide, at the same concentration, has a pH of 14. Since pH is a logarithmic scale, a difference of one in pH is equivalent to a tenfold difference in hydrogen ion concentration.
Log–log graph of the probability that a number starts with the digit(s) n, for a distribution satisfying Benford's law. The points show the exact formula, P(n) = log 10 (1 + 1/n). The graph tends towards the dashed asymptote passing through (1, log 10 e) with slope −1 in log–log scale. The example in yellow shows that the probability of a ...
Example Bjerrum plot: Change in carbonate system of seawater from ocean acidification.. A Bjerrum plot (named after Niels Bjerrum), sometimes also known as a Sillén diagram (after Lars Gunnar Sillén), or a Hägg diagram (after Gunnar Hägg) [1] is a graph of the concentrations of the different species of a polyprotic acid in a solution, as a function of pH, [2] when the solution is at ...
The ocean contains a natural buffer system to maintain a pH between 8.1 and 8.3. [11] The oceans buffer system is known as the carbonate buffer system. [ 12 ] The carbonate buffer system is a series of reactions that uses carbonate as a buffer to convert C O 2 {\displaystyle \mathrm {CO_{2}} } into bicarbonate . [ 12 ]
Because log(x) is the sum of the terms of the form log(1 + 2 −k) corresponding to those k for which the factor 1 + 2 −k was included in the product P, log(x) may be computed by simple addition, using a table of log(1 + 2 −k) for all k. Any base may be used for the logarithm table. [53]
The term grid strength (also system strength) is used to describe the resiliency of the grid to the small changes in the vicinity of the grid location (“grid stiffness”). [5] From the side of an electrical generator, the system strength is related to the changes of voltage the generator encounters on its terminals as the generator's current ...
The first such distribution found is π(N) ~ N / log(N) , where π(N) is the prime-counting function (the number of primes less than or equal to N) and log(N) is the natural logarithm of N. This means that for large enough N, the probability that a random integer not greater than N is prime is very close to 1 / log(N).