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Question
hi randy,
 when graphing on semilog graph paper, are the y values in ln or log? i've looked at a few websites and they never seem to specify, with examples in ln for some, and log for others. if i'm trying to find the slope via the graph, would it be ln-ln/t-t, or log-log/t-t? thanks.

Answer
Gary, The log axis on semi-log graph paper are usually log (base 10). One reason, I think, is because factors of 10, which can be easily seen on the graphs, correspond to scientific notation.

There is an important difference in the motivation between using semi-log and log-log graphs. For exponential functions, such as

y = f(x) = aexp(bx) where a and b are constants becomes

lny = lna + bx

so you can see that lny vs. x will look like a line on semi-log graphs. The slope of the line, b, is the "growth" constant or what ever interpretation is appropriate for a given problem. Note that we could have taken the log,

logy = loga  + log(exp(bx)) = loga + b・loge・x so that logy vs. x is a line with slope b・loge. This form would be easier to interpret on semi-log graph paper. Note also that functions like

y = ac^x can be be transformed into logy = loga + x・logc, which is also a line on semi-log paper.

Now consider a monomial function like y = ax^b. Taking the log gives

logy = loga + b・logx

so that logy vs. logx is a line on log-log paper with slope b which is the exponent of the power law represented by the function. Finding the exponent of a power law like this is very useful (eg., the slope of the spectrum of turbulence on log-log paper shows the characteristic -5/3 power law in the Kolmogorov inertial subrange, so that you can show that the phenomena you are analyzing really is turbulence).

Hope this helps.

Randy

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randy patton

Expertise

college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

Experience

26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

Publications
J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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Also an Expert in Oceanography

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