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Logarithms

 
   

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Algebra & Arithmetic

Scientific Notation

LOGARITHMS

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Common Logarithms

The common logarithm (written: log x) of a number is the power to which 10 must be raised to equal that number. The logarithm of 1000 (written log 1000) is 3 because 10 cubed (103) is equal to 1000.

1000=10 x 10 x 10 = 103

log 1000 = 3

Logarithms for numbers are generally found using a calculator since the relationship between a number and its logarithm is not generally obvious. Logarithms of numbers larger than one are positive while logarithms of numbers smaller than 1 are negative. (Logarithms are not defined for numbers less than zero.) Writing numbers in scientific notation clarifies this point as the logarithm of a number will be close in value to the value of its exponent.

log 3021 = 3.480

3021 = 3.021 x 103

log 3.021 x 103 = 3.480

For numbers larger than one, the first digit of the logarithm is equal to the exponent (when the number is written with one digit to the left of the decimal point). For numbers smaller than one, the first digit may be one less than the value of the exponent.

log 0.0000378 = -4.132

0.0000378 = 7.38 x 10-5

log 7.38 x 10-5 = -4.132

Antilogarithms

If the logarithm is known, the corresponding number is found by raising 10 to the logarithm (anti-logarithm). Suppose the logarithm of an unknown number is 3.63. The unknown number is found as follows:

log x = 3.63

10log x = 103.63

x = 4.27 x 103

Natural Logarithms

Natural logarithms (written: ln x) are identical to common logarithms, except the natural log of a number is the power to which e (2.718...) must be raised to get that number. While this number appears arbitrary it appears in many mathematical formulas.

Natural logarithms and common logarithms are related as follows:

ln x = ln 10 x log x = 2.303 log x

The natural antilogarithm of x is found by raising e to the power of x. Raising e to a power is called taking the exponential.

Logarithmic and Exponential Relationships

Some useful properties of logarithms:

Some useful properties of exponents:

Try some problems using logarithms!


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Manipulating Formulas Containing Logarithms

The general rules for manipulating formulas apply to formulas containing logarithms. At times, these formulas may seem more confusing. The appropriate application of logarithms and antilogarithms are used to isolate the variable of interest.

Example

Rearrange the following equation to find E when k, A, T and R are known.

  1. Start with the original formula:

  2. Examine the formula. What changes need to be made to obtain an expression for E in terms of the other variables?

    • Subtract ln A from both sides:

    • Multiply both sides by -1:

      NOTE: the left-hand side of the equation can be rewritten using on of the logarithmic relationships given above: .

      

    • Multiply both sides by RT:

  3. Determine the final formula:

Example

Rearrange the following equation to find A when k, E, T and R are known.

  1. Start with the original formula:

  2. Examine the formula. What changes need to be made to obtain an expression for A in terms of the other variables?

    • Add E/(RT) from both sides:

    • Now the expression is for ln A. To remove the natural log take the exponential of each side:

  3. Determine the final formula:

    The formula may be further simplified using the properties of exponents given above:

Try some problems manipulating formulas containing logarithms!


Do on-line problems!