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The ability to perform precise mathematical calculations quickly is of paramount importance throughout the course of a clinical career. On a day-to-day basis, the anesthesiologist will compute drug doses, drug concentrations, and various physiologic formulae. Even though most of the calculations should be rote, the following are more complex and close attention should be paid at each step to avoid a miscalculation that could result in patient harm.

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Basic Exponential Function

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Exponential function can be used to describe bacterial growth and radioactive decay. The “basic” exponential function is the function *y* = *a*^{x} where *a* is some positive constant called the base and *x* is the exponent. For instance, to solve the equation *y* = 4^{3} it can be expanded to *y* = 4 × 4 × 4 resulting in a *y* of 64.

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In addition to their utility in describing drug half-lives, logarithms are used in the Nernst equation describing the potential across a cell membrane and the Henderson–Hasselbalch equation governing the relationship between pH and pKa.

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A logarithm (log for short) is actually just the reverse of the exponential scale. Therefore:

log_{a}x = *y* is the same as *a*^{y} = *x*

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In the example log_{2}8 = 3, the base is the subscript number found after the letters “log (ie, 2), the argument is the number following the subscript number (ie, 8), and the answer is the number that the logarithmic expression is set equal to (ie, 3).

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*Common* logarithms (log_{10}*x*) have a base of 10. If a log is written without a base (as log *x*), then it is assumed to have a base of 10. *Natural* logarithms (ln *x*) have a base of *e* which is approximately 2.71828.

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Graphing Simple Equations

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Linear relationships can be represented in the form *y = mx + b*, where *m* is the slope of the line and *b* equals the point where the line crosses the *y*-axis (*y*-intercept).

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Points are named by an ordered pair such as (4,2) where the first number in an ordered pair is the *x*-coordinate and the second is the *y*-coordinate.

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To solve the equation and convert it into a graphical format, first select an *x*-coordinate, then solve the equation yielding a *y*-value. Repeat this process about 4 or 5 times and then connect the points you have graphed. The line you see will be the graph of a linear equation.

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For the equation *y* = 2*x* + 1:

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