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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.
With specific values for C a and K a this quadratic equation can be solved for x. Assuming [4] that pH = −log 10 [H +] the pH can be calculated as pH = −log 10 x. If the degree of dissociation is quite small, C a ≫ x and the expression simplifies to = and pH = 1 / 2 (pK a − log C a).
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 ...
where denotes the natural logarithm, with the sum extending over all prime numbers p that are less than or equal to x. The second Chebyshev function ψ ( x ) is defined similarly, with the sum extending over all prime powers not exceeding x
C A is the analytical concentration of the acid and C H is the concentration the hydrogen ion that has been added to the solution. The self-dissociation of water is ignored. A quantity in square brackets, [X], represents the concentration of the chemical substance X. It is understood that the symbol H + stands for the hydrated hydronium ion.
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]
For x > 1 let π 0 (x) = π(x) − 1 / 2 when x is a prime number, and π 0 (x) = π(x) otherwise. Bernhard Riemann, in his work On the Number of Primes Less Than a Given Magnitude, proved that π 0 (x) is equal to [9] Riemann's explicit formula using the first 200 non-trivial zeros of the zeta function
The distribution coefficient, log D, is the ratio of the sum of the concentrations of all forms of the compound (ionized plus un-ionized) in each of the two phases, one essentially always aqueous; as such, it depends on the pH of the aqueous phase, and log D = log P for non-ionizable compounds at any pH.