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Section 4.4
Logarithms are used to solve for variable exponents. For example, finding
“x” in the equation 4x = 41.
Definition 4.4- If r is any positive number, then the unique exponent t such
that b t = r is called the logarithm of r with base b and is denoted by logbr.
log b r = t
So....
log 2 8 = 3
log 3 81 = 4
2 7 = 128
is equivalent to
bt= r
is equivalent to 2 3 = 8
is equivalent to 3 4 = 81
is equivalent to log 2 128 = 7
Examples of evaluating logarithms....
log 6 36
log 6 36 = x
6 x = 36
6 x= 6 2
x=2
1) Set logarithm equal to “x.”
2) Change to exponential form.
3) Set to equal bases.
4) Use properties of exponents to evaluate
Properties of Logarithms
Property 4.3 -
For b > 0
and
b ≠ 1,
log b b = 1
Property 4.4 -
and
log b 1 = 0
For b > 0, b ≠ 1, and r > 0,
blog b r = r
Property 4.5 - For positive numbers b, r, and s, where b ≠ 1,
log b rs = log b r + log b s
Property 4.6 - For positive numbers r, r, and s, where b ≠ 1,
log b (r ⁄ s) = log b r - log b s
Property 4.7 - If r is a positive real number, b is a positive
real number other than 1, and p is any real number, then
log b r p = p(log b r)
More examples....
Solve log 5 x = 2
log 5 x = 2
5 2= x
25 = x
Express as the sum or difference of simpler logarithmic quantities.
log b xyz
= log b x + log b y + log b z
log b (∛ x2z)
(2/3) log b x + (1/3) log b z
Solve the equation: log 3 x + log 3 4 = 2
log 3 4x = 2
32 = 4x
9 = 4x
9/4 = x