8.2 The Common Logarithm Name___________________________________________________ Problem 1. Mathematical models can be used to solve various problems. Use the model s = 93log d + 65 , where s equals the speed of the wind in miles per hour and d represents the distance the tornado traveled in miles. 1. Calculate the speed of a tornado that traveled a distance of 220 miles. 2. Calculate the distance a tornado traveled if the speed was 250 miles per hour. 3. Calculate the speed of a tornado that traveled a distance of 150 miles. 4. Calculate the distance a tornado traveled if the speed was 175 miles per hour. 5. Move to page 2.2. Move point P so that its coordinates are (1, 2 ) . The point (1, 2) on f ( x ) = 2x indicates that 21 = 2 . P ′ has the coordinates ( 2, 1) . The point (2, 1) on f −1 ( x ) = log2 ( x ) indicates that log2 2 = 1 . Use this relationship between exponential expressions and logarithmic expressions to complete the following table. (Move point P as necessary.) Exponential Expression P 2 x = f(x) Logarithmic Expression P’ log 2 x = f −1 (x) (1, 2) 21 = 2 (2,1) log2 2 = 1 (2, 4) (8,3) 2 0 = 1 2 −1 =
1
2
1⎞
⎛
⎜ −2, 4 ⎟ ⎝
⎠
log2
1
= −3 8
6. Move to page 2.3. Solve the logarithmic equation log2 32 = y using the patterns from question 1. Then, use the slider to change the n-­‐value to solve the logarithmic equation. How does the exponential equation verify your result? 7. Move to page 3.1. Solve the equation log4
1
= y . Then, use the slider to change the n-­‐value to solve the 256
logarithmic equation. How does the exponential equation verify your result? 8. Maya solved the logarithmic equation log4 16 = y . She says the answer is 4 since 4 × 4 = 16 . Is her answer correct? Why or why not? 9. Alex says that when solving a logarithmic equation in the form logb a = y , he can rewrite it as ba = y . Is this a good strategy? Why or why not? The Common Logarithm log10 1 =
log10 1000 =
log10 10 =
log10
=4
log10 100000 =
log10
1
=
10
10. Changing between logarithmic form & exponential form: logb n = e ⇔ 1
=
100
1
log10
=
1000
log10
Logarithms with Other Bases (No calculator) log3 3 = x
log2 2 = x
log2 4 = x
log2 8 = x
log3 9 = x
log5 5 = x
1
=x
64
log4 16 = x
log5 1 = x
log4
1
=x
9
log3 81 = x
log3
log2 32 = x
log4 4 = x
1
=x
25
log5 125 = x
log5
log4 45 = x
Practice – Changing between Logarithmic & Exponential Form. Complete the chart. Logarithmic Form Exponential Form Logarithmic Form Exponential Form 11. log3 243 = 5 log9 9 = 1 −1
12. 40 = 1 ⎛ 1⎞
⎜ 4 ⎟ = 4 ⎝ ⎠
13. Special logarithmic values: logb b = __ and logb 1 = __ . VI. Practice – Evaluating logarithms (LOG WAR) 15. log1 2 0.25 = x 14. log2 64 = x 16. log1 3 27 = x 17. log8 x =
18. logx 36 = 2 1
3
19. logx 64 =
1
2
22. log0.1 = x 21. 6 ⋅log7 7 = x 20. l0log2.3 = x Change of Base Formula. Many calculators can only compute common logarithms (base 10) and natural logarithms (base e). While these two logarithms are the ones most frequently used, sometimes you might need to evaluate logarithms of other bases. To do this, the change of base formula is used. To discover this formula, some exploration is required. Evaluate the following. 23. log7 5 24. log3 7
log12 5
25. log10 5
log10 7
26. log2 10
log5 10
27. logπ 5
logπ 7
28. Which of the expressions in Exercises 1–5 share the same value? What do these expressions have in common? 29. Rewrite the expression logb n as the quotient of two common logarithms. (This is the change-­‐of-­‐base formula.) 30. log7 64 = x 31. log1 2 8 = x 32. log6 12 = x