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Engineering and Scientific Calculations
Published in David E. Clough, Steven C. Chapra, Introduction to Engineering and Scientific Computing with Python, 2023
David E. Clough, Steven C. Chapra
We can compute the natural logarithm to a given resolution by evaluating this series to the number of terms required to give convergence to that resolution. Notice that the logarithm is only defined for positive values of x.
The Communications Revolution
Published in John D. Cressler, Silicon Earth, 2017
Okay. Relaxed? Smartphone still off? Good. Recall from algebra that a logarithm is simply an exponent. Formally, if b is a positive number other than 1, then logbx (read this, “the logarithm to the base b of x”) represents the power to which b must be raised to produce x. For example, log10 100 = 2, because 102 = 10 × 10 = 100. In general, y = logbx and x = by are equivalent statements. Fine. This latter statement of x = by is more precisely referred to as “x is increasing exponentially (with y).” If y happens to be time (t) such that x = bt, then we call this process “exponential growth” (x could in principle either be increasing in time if we have x = by, or decreasing in time if we have x = b−y = 1/by, but it is still exponential growth).
Errors in algebra
Published in Breach Mark, Essential Maths for Engineering and Construction, 2017
It represents a misunderstanding of the property of logarithms. The logarithm of a number is the power to which its base has to be raised to be the same as the original number. For example, if the chosen base is 10 and the number for which we wish to find the logarithm is 1000, then to get the number 1000 we would need to raise the base to the power of three. That is 1000 = 103, therefore the logarithm (to the base of 10) of 1000 is 3. Some consequences of this definition are that:
Bulk etch rates of CR-39 nuclear track detectors over a wide range of etchant (NaOH aqueous solution + ethanol) concentrations: measurements and modeling
Published in Radiation Effects and Defects in Solids, 2020
Figure 5 shows the analysis of bulk etches rates of CR-39 with ethanol from different angles that is ln(Vb) variation with ethanol volumes. It shows a systematic natural logarithmic dependence of Vb on ethanol volumes present in NaOH/H2O etching solutions during etching. It may be noted that the volume of ethanol, added in the etching solution, is proportional to its concentration in the etching solution. It may be noted that the logarithm is a function inverse of the exponentiation.