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David Terr
Ph.D. Math, UC Berkeley

 

Home >> Pre-Calculus >> 3. Exponential and Logarithmic Functions

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3.3. Properties of Logarithms

Since logarithms are the inverse of exponential functions, they enjoy analogous properties which can be derived using the definition of the inverse. These properties are as follows:

  • (3.3.1) logb(xy) = logbx + logby
  • (3.3.2) logb(1/x) = -logbx
  • (3.3.3) logb(x/y) = logbx - logby
  • (3.3.4) logb(xy) = y logbx
  • (3.3.5) logcx = logbx / logbc

Logarithms satisfy the following additional properties, which follow immediately from the definition of the inverse.

  • (3.3.6) logb(bx) = x
  • (3.3.7) equation 3.3.6

Before the advent of scientific calculators, math and science students made elaborate use of these equations along with log tables in order to solve complicated equations, such as multiplication and exponentiation of large numbers. Although this practice has since by-in-large died, it is nevertheless worthwhile to get a feel for how this was accomplished.

 

Example 1: Use the log table from the previous section to estimate the product 17 * 35.

Solution: By Identity (3.3.1) we see that log (17*35) = log 17 + log 35 = 1.230 + 1.544 = 2.774. Thus, we need to find the number whose logarithm is 2.744, or in other words, to evaluate 102.774. But we know that 102.774 = 102 * 100.774 = 100 * 100.774, so it suffices to look for the number whose logarithm is 0.774. From the table, we see that log 6 = 0.778, which is just slightly larger than 0.774. Thus we conclude that 17 * 35 ≈ 100 * 6 = 600. This is very close to the actual value of 595.

 

Example 2: Use the log table from the previous section to estimate √12.

Solution: We know from Section P.4 that √12 = 121/2. Thus we have log √12 = log(121/2) = 1/2 * log 12 = 1/2 * 1.079 = 0.5395. Since log 3.5 = 0.544 is close to this value, we have √12 ≈ 3.5, which is very close to the actual value of 3.464.

 

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