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EXPONENT RULES & PRACTICE 1. PRODUCT RULE: To multiply when two bases are the same, write the base and ADD the exponents. Examples: A. B. C. 2. QUOTIENT RULE: To divide when two bases are the same, write the base and SUBTRACT the exponents. Examples: A. B. ˘ C. ˇ ˇ 3.
Properties of Exponents and Logarithms. Exponents. Let a and b be real numbers and m and n be integers. Then the following properties of exponents hold, provided that all of the expressions appearing in a particular equation are de ned. 1. aman= a+2. ( am)n= amn3. ( ab )m= a b 4. am. an. = am n, a 6= 0 5. a b m. = am. bm. , b 6= 0 6. am= 1 am.
we need to have some understanding of the way in which logs and exponentials work. De nition: If x and b are positive numbers and b 6= 1 then the logarithm of x to the base b is the power to which b must be raised to equal x.
1 EXPONENTS AND LOGARITHMS WHAT YOU NEED TO KNOW • The rules of exponents: • ma m× an = a +n • a a a m n = mn • (am)n mn= a • aa m n n m • a a n n − = 1 • an × nb = (ab)n • a b a b n n n = ⎛ ⎝⎜ ⎛ ⎝ ⎞ ⎠⎟ ⎞ ⎠ • The relationship between exponents and logarithms: • ab=⇔b xb g a where a is called the ...
Example 1: Use Logarithm Rules Expand using logarithm rules until no more can be applied. We see that the argument is first and foremost a product. Therefore, we will use the Product Rule first. There are also some exponents. Please observe that can be written as . Therefore, we can write the following:
Write 2 y = x in logarithmic notation. Note: by = x is equivalent to b(x) = y. 2 (x) = y. 2) ( ). Note: 8 = 23. 2(8) = 3. 3) Evaluate ( ). Note: (x) = (x). Thus, ( 5) = 5. 4) Rewrite ( ∙ ) as a sum, difference, or product of logarithms, and simplify if possible. 3(x6∙z2) = 3(x6) + 3(z2) = 6 ∙ 3(x) + 2 ∙ 3(z). Thus, 3(x6∙z2) = 6 ∙ 3(x) + 2 ∙ 3(z).
There are three laws of logarithms which you must know. log a x + log a y = log a ( xy ) where a , x , y > 0 . If two logarithmic terms with the same base number (a above) are being added together, then the terms can be combined by multiplying the arguments (x and y above). log 5 4 ) 2 × 4 = log 5 8. where a , x , y > 0 .
