1. Math Keeper 27
Logarithms
Essential Question – How is a log function related to an exponential function?
You use log functions to solve exponential problems; they are inverses of each other.

2. When will I use this? Human memory Intensity of sound (decibels)
Finance
Richter scale

3. Evaluating Log Expressions
We know 22 = 4 and 23 = 8
But for what value of y does 2y = 6?
Because 22<6<23 you would expect the answer to be between 2 & 3.
To answer this question exactly, mathematicians defined logarithms.
Logarithms are the INVERSE of exponential functions.

4. Definition of Logarithm to base b
Let b & x be positive numbers, b ≠ 1.
logby = x iff bx = y
This expression is read “log base b of y”
The definition tell you that the equations logby = x and bx = y are equivalent.

5. Example 1: Rewrite the equation in exponential form
Log form Exp. form
a) log232= 5
b) log51 = 0
c)log101 = 1
d) Log1/2 2 = -1
25 = 32
50 = 1
101 = 1
(1/2)-1 = 2

6. YOUR TURN!!
Log form Exp. form
e) log39= 2
f) log81 = 0
g)log5(/25)=-2
32 = 9
80 = 1
5-2 = 1/25

7. Rewriting forms:
To evaluate log3 9 = x ask yourself…
“3 to what power is 9?”
3x = 9 → 32 = 9 so…… log39 = 2

8. Example 2: Evaluate the expression without a calculator
log381
b) log50.04
c) log5125 4 -2 3

9. 4 4x = 256 4x = 64 (1/4)x = 256 2x = (1/32) d) log4256 e) log464
YOUR TURN!! 4
4x = 256
4x = 64
(1/4)x = 256
2x = (1/32)
d) log4256
e) log464
f) log1/4256
g) log2(1/32) 3 -4 -5

10. You should learn the following general forms!!!
Log b 1 = 0 because b0 = 1
Log b b = 1 because b1 = b
Log b bx = x because bx = bx

11. Natural logarithms log e x = ln x
The natural log is the inverse of the natural base, e.
ln means log base e

12. Common logarithms log 10 x = log x
Understood base 10 if nothing is there.

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

14. Example 3: Use a calculator to evaluate the expression
Example 3: Use a calculator to evaluate the expression. Round answer to 3 decimal places.
log 5
b) ln 0.1
c) log 7
d) ln 0.25
0.6989= 0.700
-2.303
0.845
-1.386

15. INVERSE PROPERTIES
g(x) = log b x is the inverse of the exponential function f(x) = bx
f(g(x)) = blogbx = x
g(f(x)) = logbbx = x
*Exponential and log functions are inverses and “undo” each other

16. Example 4: Using inverses→ Simplify the expression.2
a) 10log2 =
b) log39x =
c) 10logx =
d) log5125x =
log332x= 2x
log3(32)x = x 3x
log5(53)x =
log553x =

17. Finding Inverses Find the inverse of y = log3x
By definition of logarithm, the inverse is y=3x
OR write it in exponential form and switch the x & y! 3y = x → 3x = y

18. Example 5: Find the inverse of...
a) y = ln (x +1)
X = ln (y + 1) Switch the x & y
x = loge(y + 1) Write in log form
ex = y Write in exp form
ex – 1 = y Solve for y
y = ex – Final Answer

19. Example 5: Find the inverse of...
b) y = log8x
8y = x Switch x & y
8x = y Solve for y
y = 8x Final Answer

20. Example 5: Find the inverse of...
y = ex + 3
y = (2/5)x
y = ex + 10
c) y = ln (x - 3)
d) y = log2/5x
e) Y = ln (x–10)