Aim: How do find the logb a?
Do Now:
Aim: Evaluating Logs
Course: Alg. 2 & Trig.
Special Log Values/Properties
Let a and x be positive real numbers
such that a  1.
1. loga 1 = 0 because a0 = 1 log4 1 = 0
2. loga a = 1
because a1 = a
log4 4 = 1
3. loga ax = x because ax = ax
log x
4. a a  x

Inverse
Property
because y = ax
log4 43 = 3
3 log 3 81  81
y  log x = y
inverse  x = a
a
substitute loga x for y in x = ay
x  a log a x
Aim: Evaluating Logs
Course: Alg. 2 & Trig.
Converting Logs and Exponents
Rewrite the exponential and logarithmic equations
logarithmic
y = logb x
Equivalent Equations
exponential
by = x
log2 16 = 4
24 = 16
log3 1 = 0
30 = 1
log2 6  2.585
22.585  6
log10 10 = 1
101 = 10
log10 0.1 = -1
10-1 = 0.1
log16 4096 = 3
163 = 4096
log3 1/27 = -3
3-3 = 1/27
log2 1/8 = -3
2-3 = 1/8
Aim: Evaluating Logs
Course: Alg. 2 & Trig.
Evaluating Logs
Evaluate log8 16
Let x = log8 16
16 =
8x
24 = (23)x
Find the exponent
that makes this
statement true
Rewrite log8 16 into exponential
form in order to evaluate.
Write both sides with base 2
24 = 23x
4 = 3x
Set exponents equal to each other
4/3 = x
Solve for x
log8 16 = 4/3
Aim: Evaluating Logs
Course: Alg. 2 & Trig.
Evaluating Logs
Evaluate log7 1/49
Let x = log7 1/49
1/49 =
7x
49-1 = 7x
Rewrite log7 1/49 into exponential
form in order to evaluate.
Write both sides with base 7
(72)-1 = 7x
7-2 = 7x
Set exponents equal to each other
-2 = x
Solve for x
log7 1/49 = -2
Aim: Evaluating Logs
Course: Alg. 2 & Trig.
Evaluating Logs (con’t)
If log N = 0.6884, what is the value of N?
What do I know?
• common log - base 10
• exponent is 0.6884
• log N = 0.6884 equivalent to
100.6884 = N
N = 4.879977 . . .
Find the value of N to the thousandths
place in each of the following:
log N = 3.9394
log N = -1.7799
If 103.7924 = a, find log a
Aim: Evaluating Logs
Course: Alg. 2 & Trig.
Using Calculator to Find Value of Log10
The logarithmic function with base 10
is called the common log function.
If no subscript for base is given
assume a base 10
log 100 = 2
Find log 79
= 1.897627091 . . .
From home screen hit LOG key
and enter 79. Close parentheses
and hit ENTER .
Find log 243 = 2.385606274 . . .
Find log .384 = -.415668 . . .
Find log 343
Aim: Evaluating Logs
= 4.5944 . . .
Course: Alg. 2 & Trig.
Finding Common Logarithms
Use your calculator to find to the nearest
n log 7.83
10,000th.
0.8938
Log 7.83
=
Log 78.3
=
1.8938
characteristic
mantissa
=
3.8938
Log 783000 =
5.8938
Log 7830
If 1 < a < 10, then 0 < log a < 1 and
Log (a x 10n) = log a + n
Find log 120
= log 1.2 + log 100
120 = 1.2 x 102
Aim: Evaluating Logs
0.0792 +
2 = 2.0792
Course: Alg. 2 & Trig.
Natural Logarithmic Function
f(x) = logex = ln x,
x>0
1. ln 1 = 0 because e0 = 1
2. ln e = 1
because e1 = e
3. ln ex = x because ex = ex
4. e ln x  x
inverse property
5. If ln x = ln y, then x = y
4

v x  = e x
2
-5
The logarithmic
function with base e
is called the natural
log function.
Aim: Evaluating Logs
5
-2
ux = ln x
-4
Course: Alg. 2 & Trig.
Using Properties of Natural Logarithms
Rewrite each expression:
1
1. ln  ln e 1
e
= -1 e ln x  x inverse property
 x inverse property
e2
=2
e
3. ln e0
=0
ln ex = x because ex = ex
4. 2ln e
=2
ln e = 1 because e1 = e
2. ln
Aim: Evaluating Logs
ln x
Course: Alg. 2 & Trig.
Solving Equations w/logs
Alternate method for solving exponential equations
Solve 4x = 128
10? = 4
10? = 128
Convert each side
of equation to
power with base 10
log 4 = 0.60206
log 128 = 2.10721
(100.60206)x = 102.10721
Substitute
0.60206x = 2.10721
bx = b y  x = y
2.10721
x
 3.5
0.60206
Solve
Solve 6x = 280 to nearest thousandth
Aim: Evaluating Logs
Course: Alg. 2 & Trig.
Using Logs to Solve Problems
On June 15, 1985, Ted Nugent and the Bad
Company played at the Polaris Amphitheater
in Columbus, Ohio. Several miles away, the
intensity of the music at the concert registered
66.6 decibels. How many times the minimum
intensity of sound detectable by the human
ear was this sound, if I0 is defined to be 1?
I
Use the formula for Loudness L  10log
I0
I
66.6  10log
I0
6.66  log I
106.66 = I
Divide by 10, I0 = 1
Rewrite in exponential terms
x = 4,570,882 times
Aim: Evaluating Logs
Use the 10x key
Of your calculator
Course: Alg. 2 & Trig.