Aim: Evaluating Logs Course: Alg. 2 & Trig. Aim: How do find the log b a? Do Now:

Aim: Evaluating Logs Course: Alg. 2 & Trig.

Aim: How do find the logb a?

Do Now:


Special Log Values/Properties

Let a and x be positive real numberssuch that a 1.

1. loga 1 = 0

2. loga a = 1

3. loga ax = x

because a0 = 1

because a1 = a

because ax = ax log4 43 = 3

log4 4 = 1

log4 1 = 0

4.

a log a x x

3log 3 81 81Inverse Property

loga x = y

x a log a x

because y = ax

x = ay inverse substitute loga x for y in x = ay


Converting Logs and Exponents

log2 16 = 4

Rewrite the exponential and logarithmic equations

log3 1 = 0

log2 6 2.585

24 = 16

101 = 10

10-1 = 0.1

163 = 4096

logarithmic exponentialy = logb x by = xEquivalent Equations

30 = 1

22.585 6

log10 10 = 1

log10 0.1 = -1

log16 4096 = 3

3-3 = 1/27

2-3 = 1/8

log3 1/27 = -3

log2 1/8 = -3


16 = 8x

24 = (23)x

Evaluating Logs

Rewrite log8 16 into exponential form in order to evaluate.

Evaluate log8 16

Let x = log8 16

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

Find the exponent that makes this statement true


1/49 = 7x

Evaluating Logs

Rewrite log7 1/49 into exponential form in order to evaluate.

Evaluate log7 1/49

Let x = log7 1/49

49-1 = 7x 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


Evaluating Logs (con’t)

If log N = 0.6884, what is the value of N?

• exponent is 0.6884

• common log - base 10

What do I know?

• log N = 0.6884 equivalent to100.6884 = NN = 4.879977 . . .

Find the value of N to the thousandthsplace in each of the following:

log N = 3.9394 log N = -1.7799

If 103.7924 = a, find log a


Using Calculator to Find Value of Log10

Find log 79

From home screen hit LOG key and enter 79. Close parentheses and hit ENTER .

= 1.897627091 . . .

= 2.385606274 . . .

= -.415668 . . .

Find log 243

Find log .384

Find log 343 = 4.5944 . . .

The logarithmic function with base 10is called the common log function.

If no subscript for base is givenassume a base 10

log 100 = 2


Finding Common Logarithms

Use your calculator to find to the nearest 10,000th.

Log 7.83 =

Log 78.3 =

Log 7830 =

Log 783000 =

0.8938

1.8938

3.8938

5.8938

n

If 1 < a < 10, then 0 < log a < 1 andLog (a x 10n) = log a + n

Find log 120

120 = 1.2 x 102

= log 1.2 + log 100

0.0792 + 2 = 2.0792

log 7.83

characteristic mantissa


Natural Logarithmic Function

f(x) = logex = ln x, x > 01. ln 1 = 0

2. ln e = 1

3. ln ex = x

because e0 = 1

because e1 = e

because ex = ex

4

2

-2

-4

-5 5

u x = lnx

4

2

-2

-4

-5 5

v x = ex

The logarithmic function with base eis called the natural

log function.

4. e ln x x inverse property5. If ln x = ln y, then x = y


Using Properties of Natural Logarithms

2. ln e2

3. ln e0

xe x inverse propertyln

Rewrite each expression:

e

11. ln

xe x inverse propertyln

ln ex = x because ex = ex

= -1

= 2

= 0

4. 2ln e = 2 ln e = 1 because e1 = e

e 1ln


Solve

bx = by x = y

Substitute

Solving Equations w/logs

Solve 4x = 128Convert each side of equation to power with base 10

10? = 410? = 128

Alternate method for solving exponential equations

log 4 = 0.60206log 128 = 2.10721

(100.60206)x = 102.10721

0.60206x = 2.10721

3.5x2.10721

0.60206

Solve 6x = 280 to nearest thousandth


66.6 10logI

I0

6.66 log I

Using Logs to Solve Problems

On June 15, 1985, Ted Nugent and the BadCompany played at the Polaris Amphitheaterin Columbus, Ohio. Several miles away, theintensity of the music at the concert registered66.6 decibels. How many times the minimumintensity of sound detectable by the humanear was this sound, if I0 is defined to be 1?

Use the formula for

Loudness L 10logI

I0

Divide by 10, I0 = 1

106.66 = I Rewrite in exponential terms

x = 4,570,882 times Use the 10x keyOf your calculator

Documents

Aim: Evaluating Logs Course: Alg. 2 & Trig. Aim: How do find the log b a? Do Now: