Download AA wTrig 8.5 and 8.7 Notes (Common and Natural Glex

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
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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
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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
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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→
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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
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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