INTRODUCTION
TO LOGARITHMS
5
WHAT YOU SHOULD LEARN:
I can convert logarithmic expressions to
exponential expressions and vice versa.
I can evaluate logarithmic functions.
2.4 I can convert logarithmic expressions to
exponential expressions and vice versa.
WHAT IS A LOGARITHM?
Definition of a Logarithm
If b > 0, b ≠ 1, and x > 0, then
Logarithmic Form
Form
log b x = y
base
Exponential
iff
exponent
by = x
base
exponent
Remember: A logarithm is an
exponent!
CONVERT EACH LOGARITHMIC
EXPRESSION TO AN EQUIVALENT
EXPONENTIAL EXPRESSION.
1
① Log381 = 4
③ Log5
=
-3
125
34 = 81
5-3 =
1/125
① Log2
= -3
④ Log273 = 1
2-3 = 1/8
② Log10100 = 2
271/3 = 3
1
8
3
102 = 100
CONVERT EACH EXPONENTIAL
EXPRESSION TO AN EQUIVALENT
LOGARITHMIC EXPRESSION.
① 92 = 81
④ 102 =100
Log981 = 2
log10100 = 2
1
2
5 = 5
② 54 = 625
⑤
Log5625 = 4
log5√5 = 1/2
③
12-2
1
=
144
log121/144 = -2
THE COMMON LOGARITHM
A logarithm with base 10 or log10 is called
a common logarithm.
The common logarithm is often written
without the base.
y = logx iff
y
10
=x
THE NATURAL LOGARITHM
RECALL: E ≈ 2.71828…
A logarithm with base e or loge is called a
natural logarithm.
The natural logarithm is often written
without the base.
y = lnx
iff
y
e
=x
CONVERT EACH EXPONENTIAL
EXPRESSION TO AN EQUIVALENT
LOGARITHMIC EXPRESSION OR VICE
VERSA.
① Log x = 2
102 = x
② Ln 20.0855… ≈ 3
③ e4 ≈ x
e3 ≈ 20.0855…
Ln x = 4
④ 106 = 1,000,000
Log 1,000,000 = 6
EVALUATE THE FOLLOWING:
①log
= ①log
x19x = 361
x = 2
x = -2
30 = x
x = 1
①Log
x
①log x/
/
(1/7) = 1/49
=x
x = 2
/9-2 = x
x/ = 1/81
x3 = 216
①Logxx = 6
=3
X-1/4 = 1/2
x = 16
2.5 I can evaluate logarithmic
functions in a real world scenario.
A solution’s pH is given by the function
p(t)
log(t), where t is the
hydronium ion concentration, in moles
per liter. A sample of coffee has a pH
of
5.0. What is the approximate
hydronium ion concentration of the
sample?
The wind speed s (in miles per hour) near the
center of a tornado is related to the distance d
(in miles) the tornado travels by the equation
s = 93logd + 65.
a. On March 18, 1925, a tornado whose wind
speed was about 280 miles per hour struck
the Midwest. How far did the tornado travel?