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PRECALCULUS I
PROPERTIES OF LOGARITHMS
Dr. Claude S. MooreDanville Community
College
CHANGE OF BASE
for positive real numbers a, b, and x such that
a 1 and b 1.
alog
xlogxlog
b
ba
PROPERTIES OF LOGARITHMS
1. log b (uv) = log b u + log b v.
2. log b (u/v) = log b u - log b v.
3. log b un = n log b u.
PROPERTIES OF NATURAL LOGARITHMS
1. ln (uv) = ln u + ln v.
2. ln (u/v) = ln u - ln v.
3. ln un = n ln u.
EXPMPLE:CHANGE OF BASE
4log
1
4log
10log10log
10
10
104
EXAMPLE: EXPAND LOG EXPRESSION
)5xln()31(xln
)5xln(xln5x
xln
2
3123 2
EXAMPLE: CONDENSE LOG EXPRESSIONS
522
522
2
)y3)(yln(
)y3ln(yln
)y3ln(5yln2
EXAMPLE: Find the exact value of log 8 2 .2
.2/132/3,
2)2(2
822222
82222log
3323
232/18
yoryThus
iffy
yy
y
y
LET US WORK EXAMPLES.