8.4 Logarithmsp. 486

The inverse of an exponential function is a logarithmic function.

Logarithmic Function

x = log b y

read: “x equals log base b of y”

y = bx x = logby

These two equations are equivalent

We can convert exponential equations to logarithmic equations and vice versa, using this:

y = bx x = logby

Exponential form

logarithmic form

If y = bx

then x = logby

where y > 0

b > 0

b 1

technical stuff

base base

unknown

Another way to “read” logs:

“What is the exponent of b that gives you y?”

3log 5y

2 7loga

logba d

53 y

27a

a db

Convert to exponential form

1)

2)

3)

“What is the exponent of 3 that gives you 5?”

2 8x2

log 8x1

4y

3

100010

1 log 4y

103 log 1000

Convert to logarithmic form

4)

5)

6)

Now that we can convert between the two forms we can simplify logarithmic expressions.

Simplify

7) log2 32

8) log3 27

9) log4 2

10) log3 1

2? = 32

3? = 27

4? = 2

3? = 1

? = 5

? = 3

? = 0.5

? = 0

“What is the exponent of that gives you 32?”

“What is the exponent of 3 that gives you 27?”

Evaluate

6

1) log

36g

Common Logarithm

A common logarithm is a logarithm that is base 10.

•We like base 10 because we can evaluate it in our calculator. (Use the LOG button)

•When a logarithm is base 10, we don’t write the base. log10 = log

Common logs and natural logs with a

calculator

log10 button

ln button

Evaluate with a calculator

11) log10 10

12) 2 log10 2.5

13) log10 (-2)

Remember this means 10? = -2

= 1

= 0.7959

no solution

Try these using your calculators:

1.10x = 85

2.10x = 1.498

3.10x = -5.5

Natural Exponential Function

y = ex

Natural Base

ln e = 1

Natural Logarithmic Function

y = ex x = loge y

x = ln y

Convert to natural logarithmic form:

a. 10 = ex b. 14 = e 2x

Convert to natural exponential form:

a. ln 4 = 1.386…

b. ln 6 = 1.792…

Evaluate: ln x

a. x = 2 b. x = ½ c. x = -1

.693 -0.693 undefined

1.) ex+7 = 98

2.) 4e3x-5 = 72

3.) ln x3 - 5 = 1