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At 25 °C (77 °F), solutions of which the pH is less than 7 are acidic, and solutions of which the pH is greater than 7 are basic. Solutions with a pH of 7 at 25 °C are neutral (i.e. have the same concentration of H + ions as OH − ions, i.e. the same as pure water). The neutral value of the pH depends on the temperature and is lower than 7 ...
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 = 9,808,357 + 0.09543. We can then get 10 9,808,357 × 10 0.09543 ≈ 1.25 × ...
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.
Graph of the number of primes ending in 1, 3, 7, and 9 up to n for n < 10 000. Another example is the distribution of the last digit of prime numbers. Except for 2 and 5, all prime numbers end in 1, 3, 7, or 9. Dirichlet's theorem states that asymptotically, 25% of all primes end in each of these four digits.
An LNS can be considered as a floating-point number with the significand being always equal to 1 and a non-integer exponent. This formulation simplifies the operations of multiplication, division, powers and roots, since they are reduced down to addition, subtraction, multiplication, and division, respectively.
K 1, K 2 and DIC each have units of a concentration, e.g. mol/L. A Bjerrum plot is obtained by using these three equations to plot these three species against pH = −log 10 [H +] eq, for given K 1, K 2 and DIC. The fractions in these equations give the three species' relative proportions, and so if DIC is unknown, or the actual concentrations ...
Logarithmic dilution. A serial dilution is the step-wise dilution of a substance in solution, either by using a constant dilution factor, or by using a variable factor between dilutions. If the dilution factor at each step is constant, this results in a geometric progression of the concentration in a logarithmic fashion.
[10]: 280–4 Hence, a single experiment can be used to measure the logarithms of the partition coefficient (log P) giving the distribution of molecules that are primarily neutral in charge, as well as the distribution coefficient (log D) of all forms of the molecule over a pH range, e.g., between 2 and 12.