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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