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Logarithms:

A.  Logarithms, usually called logs, can be a little complicated since there are so many properties to learn.  To become good at the log problems requires lots of practice.

B.  There are 2 basic logs:  log_n and log_e (which is written as ln and means "natural log").

1.  In general, log_b x = n means bⁿ = x and ln x = n means, eⁿ = x. 

2.  If no 'b' is given, then it is assumed the value 10.

Ex [1]  log 1000 = _______

a.  Since no value 'b' is given, b = 10.

b.  The problem is 10^x = 1000.  For this to be true, x = 3.

c.  The answer is 3.

Ex [2]  log₄ x = ³/₂, then x = _____

a.  The answer is 4^3/2 = x or 8 = x.  The answer is 8.

Ex [3]  log_b 81 = log₂ 16, then b = ______

a.  This means b^log2¹⁶ = 81.  So the first thing to do is to find log₂ 16 which means 2^x = 16.  So x = 4.  So log₂ 16 = 4.

b.  So substituting we get b⁴ = 81.  So b = 3.

c.  The answer is 3.

C.  Properties Of Logs:

1.  Below are the common properties of logs.  All of these should be memorized and you should be very familiar with knowing how to manipulate them.

2.  There might be other properties, but these are the ones that show up on the number sense tests the most.

D.  Examples Taken From Past UIL Tests

Ex [4]  log₄ 27 log₄ 3 = _________

a.  In this problem, if we can make the 1^st term be log₄ 3 then the problem would be easy.

b.  Change log₄ 27 to log₄ 3³ = 3*log₄ 3.

c.  So now we have 3 x log₄ 3 log₄ 3 = 3 x 1 = 3.

d.  The answer is 3.

Ex [5]  ln 5 + ln 8 = ln 10 - ln x, then x = ______

a.  ln 5 + ln 8 = ln 5*8 = ln 40.

b.  ln 10 - ln x = ln ¹⁰/_x.

c.  So we have ln 40 = ln ¹⁰/_x.  The only way for this to be true is if 40 = ¹⁰/_x.  So, solving for x we get x = ¹/₄. 

d.  The answer is ¹/₄.

Ex [6]  (log₂ 7)(log₇ 4) = _______

 a.  We know that (log_b a) (log_a b) = 1.  So we need to change the 2^nd term to be log₇ 2 to use this property.

b.  Change log₇ 4 to log₇ 2² = 2*log₇ 2.

c.  So we have 2 x log₂ 7 x log₇ 2 = 2 x 1 = 2.

d.  The answer is 2.

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