1. 8.4 Logarithms
p. 486

2. Daily Question:
How do you evaluate
a logarithm?

3. Logarithmic Function x = log b y read: “x equals log base b of y”
The inverse of an exponential function is a logarithmic function.
x = log b y
read: “x equals log base b of y”

4. These two equations are equivalent
We can convert exponential equations to logarithmic equations and vice versa, using this:
y = bx
x = logby
These two equations are equivalent

5. y = bx
x = logby
Exponential form
logarithmic form

6. If y = bx then x = logby Another way to “read” logs:
unknown
If y = bx
then x = logby
base
base
Another way to “read” logs:
“What is the exponent of b that gives you y?”
where y > 0
b > 0
b  1
technical stuff

7. Convert to exponential form
“What is the exponent of 3 that gives you 5?” 1) 2) 3)

8. Convert to logarithmic form4) 5) 6)

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

10. Simplify “What is the exponent of that gives you 32?”
7) log2 32
2? = 32
? = 5
“What is the exponent of 3 that gives you 27?”
8) log3 27
3? = 27
? = 3
9) log4 2
4? = 2
? = 0.5
10) log3 1
3? = 1
? = 0

11. Evaluate
try:

12. Common Logarithm A common logarithm is a logarithm that is base 10.
When a logarithm is base 10, we don’t write the base.
log10 = log
We like base 10 because we can evaluate it in our calculator. (Use the LOG button)

13. Common logs and natural logs with a calculator
log10 button
ln button

14. Evaluate with a calculator
= 1
11) log10 10 =
12) 2 log10 2.5
no solution
13) log10 (-2)
Remember this means 10? = -2

15. Try these using your calculators:
10x = 85
10x = 1.498
10x = -5.5

16. Natural Exponential Function
y = ex
Natural Base
ln e = 1

17. y = ex x = loge y x = ln y Natural Logarithmic Function
the natural logarithmic function
is the inverse of the
natural exponential function

18. 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…

19. Evaluate: ln x
a. x = b. x = ½ c. x = -1
.693
-0.693
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20. Solving natural bases
1.) ex+7 = 98
2.) 4e3x-5 = 72
3.) ln x3 - 5 = 1

21. page 490
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Homework