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.