College Algebra: Lesson 7.4 Properties and Applications of Logarithms

·  Since a logarithmic statement can be written as an exponential statement, it should not be surprising that the properties of logarithms are based on the properties of exponents. The properties of logarithms allow us to change the form of logarithmic statements so that the products can be converted to sums, quotients can be converted to differences, and powers can be converted to products.

Properties of Logarithms

For x > 0, y > 0, a > 0, and any real number r:
Product property: or

Quotient property: or

Power property: or

Examples: Use the properties of logarithms to expand each expression.

1. 2. 3.

4. 5. 6.

7.

Examples: Use the properties of logarithms to condense each logarithmic expression.

1. 2. 3.

4.

Change of Base

The calculator will only evaluate common logs (base 10) and natural logs (base e). These are most commonly used in problems, but occasionally you will have a logarithm with a different base. We can use the change of base formula to express the logarithm using either the common logarithm or the natural logarithm.

Formula: Let a, b, and x be positive real numbers such that and . Then can be converted to a different base using one of the following formulas.

1. 2. 3.

Ex: Evaluate using the change of base formula with common logs and natural logs.

1. 2.

3. Evaluate the logarithmic expression. Round to 2 decimal places.

4. Use the properties of logarithms to write the following expression as a single term that doesn't contain a logarithm.

5. If a sample of a certain solution is determined to have a concentration of moles/liter, what is its pH? Round to one decimal place. (Note: the pH of a solution is defined to be , where is the concentration of hydronium ions in moles/liter.)

6. Given that watts/meter2, what is the intensity of a sound for which the decibel level of the sound measures 127? Round to 3 decimal places. (Note: In the decibel formula, , D is the decibel level, I is the intensity of the sound being analyzed, and is the intensity of a just-discernable sound.)