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Advanced Algebra w/Trigonometry Notes Section 8.5 and 8.7 – Common Logarithms/Natural Logarithms Base 10 & Base e Target Goals: 1. Find common and natural logarithms and antilogarithms 2. Evaluating logarithmic expressions using the change of base formula 3. Evaluate expressions involving the natural base and natural logarithms Review: Solve each logarithmic equation 1. log 2 6 log 2 (3 x) log 2 48 2. 2log3 x log3 ( x 2) 2 _________________________________________________________________________________________ Base 10 logarithms are called Common Logarithms. These are usually written without the subscript 10, so log10 x is written log x . The calculator can be use to find common logarithms!! Sometimes an application of logarithms requires that you use the inverse of logarithms, or antilogarithms. The calculator can be used to find antilogarithms. Use your calculator and find each of the following. 1. log 286.1 2. log .0048 4. anti log 2.162 5. anti log 1.42 3. log 6.15 6. anti log 4.111 __________________________________________________________________________________________ It is possible to evaluate expressions involving logarithms with different bases. Since your calculator isn’t programmed with all of the possible bases for logarithms, the change of base formula is very helpful. Change of Base Formula: For all positive numbers a, b and n, where a 1 and b 1, log a n logb n logb a __________________________________________________________________________________________ Express each logarithm in terms of common logarithms. Then find its value. Example: log 4 22 log 22 2.2295 by the change of base formula. log 4 Use your calculator to approximate each of the following to four decimal places using the change of base formula 7. log5 15 8. log 4 100 9. log15 5 Natural Logarithms→ __________________________________________________________________________________________ The number e, used in science and math, is an irrational number whose value is approximately 2.718, is the base for the Natural Logarithms, which is abbreviated ln. You can take antilogarithms of natural logarithms as well. The symbol for the antilogarithm of x is anti ln x . Use your calculator to find each of the following. 10. ln 732 11. ln1685 12. ln 0.0824 13. anti ln1.3475 15. anti ln3.111 14. anti ln 0.0813 _________________________________________________________________________________________ You can write an equivalent base e exponential equation for a natural logarithmic equation by using the fact that ln 4 x loge 4 x e x 4 Write each exponential equation in logarithmic form. 16. e x 8 17. e5 x Write each logarithmic equation in exponential form. Then solve for x. 18. ln x 0.7774 HW #5: Worksheet #5 19. ln10 x