1. Section 9.6 There are two logarithmic bases that occur so frequently in applications that they are given special names. Common logarithms are logarithms to base 10. Natural logarithms are logarithms to base e (an irrational number which is approximately equal to 2.7183).

2. Common Logarithms log x means log 10 x You can use a calculator to approximate common logarithms using the button below. LOG To find exact values of common logarithms, use the definition of logarithms to rewrite the expressions in exponential form to evaluate.

3. Example Find the exact value of each of the following logarithms. 1)log 10,000 log 10 4 = 4 2) log 0.001 log 10 -3 = -3 log 10 ½ = ½

4. Example Solve the following equation for the variable. Give both an exact answer and an answer approximated to four decimal places. (exact answer) (approximate answer)

5. One of the most popular uses of common logarithms involves the Richter scale for measuring the intensity of earthquakes. For R (magnitude of the earthquake), a (amplitude in micrometers of the vertical motion of the ground at the recording station), T (number of seconds between successive seismic waves), and B (adjustment factor that takes into account the weakening of the seismic wave as the distance increases from the epicenter of the earthquake), the formula is...

6. Example Find the intensity R of an earthquake when the amplitude a is 300 micrometers, time T between waves is 2.5 seconds, and B is 2.6. Round the answer to one decimal place.

7. Natural Logarithms ln x means log e x You can use a calculator to approximate natural logarithms using the button below. LN To find exact values of common logarithms, use the definition of logarithms to rewrite the expressions in exponential form to evaluate.

8. Example Find the exact value of each of the following logarithms. 1)ln e 4 4 ⅓

9. Example Solve the following equation for the variable. Give both an exact answer and an answer approximated to four decimal places. (exact answer) (approximate answer)

10. Logarithms have many uses in applications. However, most calculators only have the ability to calculate common or natural logs, but not logarithms to other bases. Therefore, we need to be able to change the base of our logarithms so that we can approximate them, when necessary.

11. Change of Base If a, b, and c are positive real numbers and neither b nor c is 1, then

12. Example Approximate log 5 to four decimal places.