Semi-log plot

on the y-axis (vertical), and a linear scale on the x-axis (horizontal). Plotted lines are: y = 10 x (red), y = x (green), y = log(x) (blue). on the x axis, and a linear scale on the y axis. Plotted lines are: y = 10 x (red), y = x (green), y = log(x) (blue). In science and engineering, a semi-log plot/graph or semi-logarithmic plot/graph has one axis on a logarithmic scale, the other on a linear scale. It is useful for data with exponential relationships, where one variable covers a large range of values. All equations of the form y=\lambda a^{\gamma x} form straight lines when plotted semi-logarithmically, since taking logs of both sides gives : \log_a y = \gamma x + \log_a \lambda. This is a line with slope \gamma and \log_a \lambda vertical intercept. The logarithmic scale is usually labeled in base 10; occasionally in base 2: : \log (y) = (\gamma \log (a)) x + \log (\lambda). A log–linear (sometimes log–lin) plot has the logarithmic scale on the y-axis, and a linear scale on the x-axis; a linear–log (sometimes lin–log) is the opposite. The naming is output–input (y–x), the opposite order from (x, y). On a semi-log plot the spacing of the scale on the y-axis (or x-axis) is proportional to the logarithm of the number, not the number itself. It is equivalent to converting the y values (or x values) to their log, and plotting the data on linear scales.…

on the y-axis (vertical), and a linear scale on the x-axis (horizontal). Plotted lines are: y = 10 x  (red), y = x (green), y = log(x) (blue).]] on the x axis, and a linear scale on the y axis. Plotted lines are: y = 10 x  (red), y = x (green), y = log(x) (blue).]] In science and engineering, a semi-log plot/graph or semi-logarithmic plot/graph has one axis on a logarithmic scale, the other on a linear scale. It is useful for data with exponential relationships, where one variable covers a large range of values. All equations of the form y=\lambda a^{\gamma x} form straight lines when plotted semi-logarithmically, since taking logs of both sides gives \log_a y = \gamma x + \log_a \lambda. This is a line with slope \gamma and \log_a \lambda vertical intercept. The logarithmic scale is usually labeled in base 10; occasionally in base 2: \log (y) = (\gamma \log (a)) x + \log (\lambda). A log–linear (sometimes log–lin) plot has the logarithmic scale on the y-axis, and a linear scale on the x-axis; a linear–log (sometimes lin–log) is the opposite. The naming is output–input (y–x), the opposite order from (x, y). On a semi-log plot the spacing of the scale on the y-axis (or x-axis) is proportional to the logarithm of the number, not the number itself. It is equivalent to converting the y values (or x values) to their log, and plotting the data on linear scales. A log–log plot uses the logarithmic scale for both axes, and hence is not a semi-log plot. F(x) = m \log_{n}(x) + b. \, The equation for a line on a log–linear plot, with an ordinate axis logarithmically scaled (with a logarithmic base of n), would be: \log_{n}(F(x)) = mx + b F(x) = n^{mx + b} = (n^{mx})(n^b). m = \frac {F_1 - F_0}{\log_n (x_1 / x_0)} which leads to F_1 - F_0 = m \log_n (x_1 / x_0) or F_1 = m \log_n (x_1 / x_0) + F_0 = m \log_n (x_1) - m \log_n (x_0) + F_0 which means that F(x) = m \log_n (x) + \mathrm{constant} In other words, F is proportional to the logarithm of x times the slope of the straight line of its lin–log graph, plus a constant. Specifically, a straight line on a lin–log plot containing points (F 0 , x 0 ) and (F 1 , x 1 ) will have the function: F(x) = (F_1 - F_0) {\left[\frac{\log_n (x / x_0)}{\log_n(x_1 / x_0)}\right]} + F_0 = (F_1 - F_0) \log_{\frac{x_1}{x_0}}{\left(\frac{x}{x_0}\right)} + F_0 m = \frac {\log_n (F_1 / F_0)}{x_1 - x_0} which leads to \log_n(F_1 / F_0) = m (x_1 - x_0) Notice that n log n (F 1 ) = F 1 . Therefore, the logs can be inverted to find: \frac{F_1}{F_0} = n^{m(x_1 - x_0)} or F_1 = F_0n^{m(x_1 - x_0)} This can be generalized for any point, instead of just F 1 : F(x) = {F_0} n^{\left(\frac {x-x_0}{x_1-x_0}\right) \log_n (F_1 / F_0)}

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