Download More notes on logs: log x represents a base 10 log , i.e. log x = log

More notes on logs:
log x represents a base 10 log
, i.e. log x = log
base is written in a log statement, then the base is 10. (Sort of like the radical with no index means square root...)
ln x represents a base e log , i.e. ln x = log
represent "natural log" or the log with the "natural" base...
natural base.
10 x. Notice that if no e x. We use ln to e is that We looked at the graph of y = log
b x when b > 1. We saw that the graph had a vertical asymptote at x = 0, contained the points (1, 0) and (b, 1). The graph showed increased from quadrant IV into quadrant I. Let's look at the graph of y = log b x when 0 < b < 1. Picture the graph of the inverse exponential decay function and reflect in the line y = x, switch all the "x" and "y" stuff to get the graph of this log function. Compare and contrast the graphs of these two basic logs, the one with b > 1 and the one with 0 < b < 1. 1
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Properties of logs (because logs are exponents!!!!)
log b (mn) = log
exponenets...
examples: b (n)
because when you multiply numbers, you add their Write as the log of a single number: log 2 + log 3
log 2 + log 3 = log (2*3) = log 6
Write using logs of prime numbers only: log 24
log 24 = log (2*2*2*3) = log 2 + log 2 + log 2 + log 3 = 3log 2 + log 3
log b (m/n) = log
exponents....
examples:
b (m) + log
b (m) ­ log
b (n)
because when you divide numbers, you subtract their Write as the log of a single number: log 18 ­ log 24
log 18 ­ log 24 = log (18/24) = log (3/4)
Simplify: log 50 + log 4 ­ log 2
log 50 + log 4 ­ log 2 = log (50*4/2) = log 100 = 2
If log 2 = a and log 3 = b, write each of the following in terms of a and/or b
a) log 6: log 6 = log 2 + log 3 = a + b
b) log (3/5) = log 3 ­ log 5 = b ­ log 5 = b ­ log (10/2) = b ­ (log 10 ­ log 2) = b ­ (1 ­ a)
= b ­ 1 + a
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log b (m n ) = n*log
the exponents....
examples:
b (m) because when you take the power of a power you multiply 3
log 8 = log 2
= 3 log 2
4
log 10000 = log 10
= 4 log 10
= 4 (reinforces the inverse relationship between logs and exponentials of the x
property that log
b b = x.)
same base and the Combining the three properties:
Simplify log 12 + 2log 4 ­ 3log 8
log 12 + 2log 4 ­ 3log 8 = log 12 + log 4
= log (12*4 2 /8 3 )
= log (3/8)
If log
4 3 = m and log
4 5 = n, write log
4 3 = m and log
log
log 4 (12)
4 5 = n, write log
4 (1/12) = log
4 (4*3)
= ­1(log 4 (4*3) )
= ­1(log 4 4 + log
= ­1(1 + m)
= ­1 ­ m
­ log 8
3
4 60 in terms of m and/or n.
log 4 60 = log 4 (4*3*5) = log 4 4 + log 4 3 + log
= 1 + m + n
If log
2
­1
4 5 4 (1/12) in terms of m and/or n.
4 3)
OR
log 4 (1/12)
= log 4 (1) ­ = 0 ­ (log 4 4 + log
= ­ (1 + m)
= ­1 ­ m
4 3)
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Change of base rule. We do sometimes need to find decimal approximations for log expressions. We can estimate these values to the nearest integer but to get a better estimate we need to use a calculator. Our calculator has two log keys: log and ln. This implies we have to be able to write every log expression using base 10 or base e . We use the definition of logs to "get rid" of the base we don't want and then take the log using the base we do want to derive this rule.
Example: Consider log
5 16 1
We know log 5 16 is somewhere between 1 and 2 (because 5
= 5 and 5
for the value of log
5 16 we need to switch this expression to one using base 10 or base 2
= 25). But if we want a better approximation e . Let's switch to base e.
x
log 5 16 = x means 5
= 16 by definition of log function
ln (5 x ) = ln 16
take natural log of both sides of the equation
x ln 5 = ln 16
using the "zoom" or power rule
x = ln 16
ln 5
divide both sides by ln 5
Now we use the calculator to approximate a value for this expression.
Notice where the 16 and the 5 in the original log function are in the new expression.
Thus the change of base rule is log b m = log a m
log
ab
where a is any new base you want
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