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Logarithms and Exponentiation Guide

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Skills Refresher: Exponentiation, Scientific Notation, Logarithms, and pH Logarithms and exponentiation are opposite mathematical functions involving two numbers, one a base and the other an exponent. This method of expressing numbers is important because many relationships found in the environment and in nature are logarithmic, such as the pH scale. Exponentiation Exponentiation expresses a base number (a) to the power of an exponent (n): x=an. This is also expressed as "a raised to the nth power" (where a and n are numbers). Here are some examples for different exponents (the * symbol indicates multiplication): Exponent Equivalent Examples positive number multiplying a by itself n times 64 = 6*6*6*6 = 1,296 negative number multiplying the reciprocal of a by itself n times 4-3 = 1/43 = 1/(4*4*4) = 1/64 1 (one) multiplying a by itself one time 171 = 17 0 (zero) called an 'empty product'; result for any non-zero base always 1 270 = 1 Common Exponents Exponents of 2 (squaring the number), used in the binary number system, are important in computer work, since computers can only choose between two options, commonly expressed as 0 and 1, represented by one bit. Commonly used to describe file sizes, there are 8 bits in one byte (computers show file sizes in kilobytes, megabytes, and gigabytes). One byte can be any of 28 (=2*2*2*2*2*2*2*2 = 256) values. The result of a number with the exponent 3 is the cube of the number: a3 = a*a*a. We use a base ten (decimal) number system, and exponentiation with base 10 (called scientific notation) is commonly used to describe very large and very small numbers. For instance, the number of seconds in a day (60 sec/min x 60 min/hr x 24 hr/day = 86400 sec/day) can be expressed using exponentiation as 8.64x104 sec/day, which literally means 8.64x10x10x10x10. Notice the '864' digits remain the same no matter how it is expressed; what changes is the location of the decimal point. Going from 86400 to 8.64x104 requires moving the decimal point 4 digits to the left and adding the 104 expression. This demonstrates a quick method to work out exponents in base 10: the number of digits the decimal point moves is the exponent on the base 10; moving the decimal point left results in a positive exponent, while moving right results in a negative exponent. Remember, in scientific notation the sign (negative/positive) on the number and the exponent mean different things: 8200 = 8.2 x 103 -8200 = -8.2 x 103 0.0082 = 8.2 x 10-3 -0.0082 = -8.2 x 10-3 An exponent does not have to be a whole number. An expression like 147.54 is completely legitimate and calculable (although more difficult to calculate by hand). Math Operations Using Exponents Operation Calculation With Same Base Calculation With Different Base Addition, Subtraction Follow order of operations (see right) Follow order of operations: calculate Multiplication Add the exponents. E.g., a3*a4 = a7 exponent first, then other operation(s). 3 4 Division Subtract the 2nd exponent from the 1st. E.g., 4 +3 = 64+81 = 145; 3 2 E.g., a3/a4 = a-1 3 *4 = 27*16 = 432 Exponent of an exponent Multiply the exponents. E.g., (a3)4 = a12 (not possible) Remember to follow order of operations when doing the math: exponents and roots first, followed by multiplication and division, followed finally by addition and subtraction, with expressions inside parenthesis calculated first. A common error is to apply the exponent individually to each term in an expression under the power, as in this incorrect expression (4+9)2 = 42+92 = 97. In the correct calculation, the addition is inside parenthesis and is completed first: (4+9)2 = 132 = 169. Logarithms The logarithm of a number to a given base is the exponent (power) to which the base must be raised in order to produce that number. It is commonly expressed n=logb(x), which literally is expressed 'the logarithm of x to the base b'. This operation is the opposite of exponentiation, x=bn. continued... Exponentiation, Scientific Notation, Logarithms, and pH page 2 Common bases include 2, 10, and the natural logarithm e (2.718). Often base 10 logarithms are written without the base 10 [e.g., log(x)]. The natural logarithm "e" is an 'irrational' constant with an approximate value 2.718. It is used in several mathematical processes, and can be relevant in environmental work, including statistical analysis of data collected from a lab experiment or field investigation. Math Operations Using Logarithms Operation Calculation With Same Base Calculation With Different Base Addition Multiply the logarithms. Follow order of operations: calculate E.g., log10(4) + log10(7) = log10(4*7) = log10(28) logs first, then other operation(s). Subtraction Divide the logarithms. E.g., log10(3.2) + log2(16) = E.g., log10(4) - log10(7) = log10(4/7) = log10(0.5714) 0.5051+4 = 4.5051 multiplication and division Follow order of operations (see right) Examples of Applications for Exponents and Logarithms These functions are used for many purposes in mathematics, and there are several applications in environmental science. Exploring/Graphing Data Data can be plotted on a logarithmic scale instead of linear, as a way to show different data relationships and more clearly view large data ranges (see example at below and right). food amount (g) plant growth (g) 1 5 10 8 100 58 1000 382 Both graphs display the same data shown in the table. The graph on the left uses linear scales for both axes, while the right graph uses a logarithmic (base 10) scale for the x-axis (note the scale values of 1, 10, 100 & 1000). In the linear axis graph (left), it is difficult to see any difference between the first two data points on the x axis, while this difference is clear in the right graph. Further, it is easier to see the relationship between these two variables in the right graph because the x-axis uses a log scale. pH and Logarithms The pH of a solution indicates its acidity or bascidity (alkalinity). pH approximates p[H], which is the negative + + logarithm (base 10) of the molar concentration of the hydronium ion H3O (sometimes referred to as H , the + + hydrogen ion or proton concentration). i.e. pH = -log10[H3O ] = -log10[H ] A water molecule is composed of 2 hydrogen and one oxygen atom (H2O). But what is really happening in water is a continuous chemical reaction attempting to reach equilibrium: Water consists almost entirely of liquid H2O at any given time, + - with only a very small amount existing as H3O and OH aqueous ions. For instance, at 'neutral' pH (same amount + - + -7 of H3O and OH in the water), the concentration of H3O is 0.0000001 moles/Litre, or 1 x 10 mol/L. To make it easier to understand and express this concentration, pH is used, equal to the negative log of this concentration: -7 log10(1 x 10 ) = -7, and the negative value of -7 = 7. So the pH of neutral water is 7. Because the pH scale is a logarithmic base 10 calculation, a change of 1 pH (e.g., pH 7 to pH 6) describes a change in acid concentration of 10 times, and because it is a negative log, a lower number indicates higher acidity (e.g., pH -6 + 6 water, with 1 x 10 mol/L H3O , is ten times more acidic than pH 7 water). So if acid rain reduces the pH of a lake's water from 6 to 4, it may not sound like much, but the change is actually very significant: • 2 pH units decrease: 1 pH unit is 10 times different in concentration, so 2 units on the log scale is 10x10=100 times different in concentration, and lower pH means more acid, so pH 4 water is 100 times more acid than pH 6 water. Robert Loney, Environmental and Resource Studies Program, Trent University rev. 2010-August .
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