Logarithmic & Exponential Functions
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Exponential Functions:(Review) Rule 1: a n a m = a n + m Example: Let a = 5, n = 2, and m = 6. 5 2 * 5 6 = 25 *15625 = 390625 and 5 2+6 = 58 = 390625 Rule 2:
an = a n− m am
Example: Let a = 5, n = 2, and m = 6.
52 25 = = 0.0016 and 5 2−6 = 5 −4 = 0.0016 6 15625 5
Rule 3: (a n ) m = a nm Example: Let a = 5, n = 2, and m = 6. (5 2 ) 6 = (25) 6 = 244140625 and (5) 2*6 = (5)12 = 244140625
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Logarithmic & Exponential Functions
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Logarithmic Functions: Functions The exponential function y = 2x means that the quantity y is obtained by raising the base 2 to the power x. The power is called the logarithm of the quantity to the given base. The exponential statement y = 2x can be written as a logarithmic statement as follows: The logarithm of y to the base 2 is x, and this statement can be written as: log2y = x Similarly the exponential statement y = px, when stated in logarithmic terms becomes: logpy = x An exponential expression can be written as a logarithmic expression and vice versa. In order to do that you need to identify the quantity, the base and the power. The quantity is obtained by raising the base to the given power. Here y is the quantity, p is the base and x is the power. Therefore, the equivalent exponential statement is: y = px. Example:
Express 100 = 102 in logarithmic form.
Solution:
Here 100 is the quantity, 10 is the base and 2 is the power. Remember: logarithm
of the quantity to the given base is the power. power Therefore, we have logarithm of 100 to the base 10 is 2 which we write as:
log10100 = 2
Example:
Express log864 = 2 in exponential form.
Solution:
Here 64 is the quantity, the base is 8 and the power is 2. Therefore, it becomes: 64 = 82.
Example:
Express y + c = kt in logarithmic form.
Solution:
Here y + c is the quantity, k the base and t the power. Therefore, it becomes:
logk(y+c) = t Example:
Express ln(y + yo) = kt in exponential form and solve for y.
Solution:
Remember ln means that the base is e. Here the quantity is (y + yo), base is e and
the power is kt. Therefore, the exponential form is: y = ekt – yo Example: In the equation 14 0 = 1 , the base is 14 and the exponent is 0. Remember that a logarithm is an exponent, and the corresponding logarithmic equation is log 14 1 = 0 where the 0 is the exponent.
1 1 Example: In the equation ( ) 0 = 1 , the base is and the exponent is 0. Remember that a 2 2 logarithm is an exponent, and the corresponding logarithmic equation is log 1 1 = 0 . 2
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Logarithmic & Exponential Functions
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Example: Use the exponential equation x 0 = 1 to write a logarithmic equation. The base x is greater than 0 and the exponent is 0. The corresponding logarithmic equation is log x 1 = 0 . ࢍࢇ ࢇ = 1 because ܽଵ = ܽ Example: In the equation 31 = 3 , the base is 3, the exponent is 1, and the answer is 3. Remember that a logarithm is an exponent, and the corresponding logarithmic equation is log 3 3 = 1 . Example: In the equation 871 = 87 , the base is 87, the exponent is 1, and the answer is 87. Remember that a logarithm is an exponent, and the corresponding logarithmic equation is
log 87 87 = 1 . Example: Use the exponential equation p1 = p to write a logarithmic equation. If the base p is greater than 0, then log p p = 1 . ࢍࢇ ࢇ࢞ = ࢞ since ࢇ࢞ = ࢇ࢞ Example: Since you know that 3 4 = 3 4 , you can write the logarithmic equation with base 3 as
log 3 3 4 = 4 . Example: Since you know that 13 4 = 13 4 , you can write the logarithmic equation with base 13 as log13 13 4 = 4 . Example: Use the exponential equation 4 2 = 16 to write a logarithmic equation with base 4. You can convert the exponential equation 4 2 = 16 to the logarithmic equation log 4 16 = 2 . Since the 16 can be written as 4 2 , the equation log 4 16 = 2 can be written log 4 4 2 = 2 . Properties of logarithms: (Learn after proving) loga xn = n loga x (the power rule) loga(xy) = loga x + loga y (the product rule)
log a
x = log a x − log a y y (the quotient rule)
loga x =
ln x ln a
(change of base rule)
ࢍ࢞ (࢞࢟ ) = ࢟
Example
Let y = eloge x
Taking ln of both sides, we have: logey= loge x logee = lnex This means that y = x. Or, x=eloge x
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Logarithmic & Exponential Functions Example: Example:
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Write ln(x – 2) – ln(x + 2) as the logarithm of a single quantity.
Using the quotient rule
ln( x − 2) − ln( x + 2) = ln
x−2 x+2
Use properties of logarithm to expand ln ( x − 1)( x − 2)( x − 3)
Example: Example:
1 2
ln ( x − 1)( x − 2)( x − 3) = ln[( x − 1)( x − 2)( x − 3)] 1 = ln[( x − 1)( x − 2)( x − 3)] 2 Now we use the product rule. 1 1 ln[( x − 1)( x − 2)( x − 3)] = [ln( x − 1) + ln( x − 2) + ln( x − 3)] 2 2 Example:
Use properties of logarithm to expand
ln 3
x2 + 1 x2 − 1
1
x2 + 1 3 1 x2 + 1 1 x2 + 1 2 2 ln 3 2 = ln 2 = ln 2 = ln( x + 1) − ln( x − 1) x −1 3 x −1 3 x −1 1 = ln ln( x 2 + 1) − ln( x + 1)( x − 1) 3 1 = ln( x 2 + 1) − ln( x + 1) − ln( x − 1) 3
Example: Example: Solve for y in the equation ln(y–yo)=kt+ln c Solution: Rearranging the equation we get ln(y–yo)–ln c=kt.
ln
y − y0 = kt c
Changing this logarithmic statement to an exponential statement, we get:
y − y0 = ekt c
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y–yo=cekt
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033-40034819
Logarithmic & Exponential Functions
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Summary Rules for Exponents 10 x = 10 ×10 × 10....x times
Rules for Logarithms log10 10 x = x
Definition of log
10 x10 y = 10 x + y
log10 xy = log10 x + log10 y
Logs turn multiplication into addition
10 x = 10 x − y 10 y
log10
(10 )
= 10 x* y
log10 x y = y log10 x
10 − x =
1 10 x
10 0 = 1
1 − log( x) = log10 x log10 1 = 0
101 = 10
log10 10 = 1
x y
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x = log10 x − log10 y y
Notes:
Logs turn division into subtraction. Logs turn exponentiation into multiplication… lets you solve for exponent!!! Look a lot like the division to subtraction rule….. Any number to the zero power equals one. Any number to the first power equals itself.
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Logarithmic & Exponential Functions
EDUDIGM
Logarithm Problem Sheet 1. Find (a)
(b)
(c)
(d)
(e)
2. Prove that 3. 4. Find 5. Prove that 6. Prove that 7. Prove that 8. Prove that 9. If
, prove that
10. Prove that , then prove that
11. If 12. Prove that 13. Given
prove that
14. Prove that 15. Prove that 16. . If 17. If
, prove that , prove that
18. Prove that 19. If a,b>0 are roots of
where p,q>0 and
prove that
20. 21. If a is the hypotenuse of a right angled triangle with sides b and c, prove that the following relation holds 22. 23. If
find
24. If
, find
25. If
find
26. Find the value of integer less than or equal to x
where [x] represents the greatest
Logarithmic & Exponential Functions 27. Solve for x and y given that xy=64 and 28. Prove that 29. If
if and only if 0<x<1
30. prove that
31. Given 32. If 33. If 34. Find
find prove that
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