IMC
Unit 5
Name:_________________________________ Date:_____________
Solving for answer.
Solving for base.
Solving for exponent.
1)
34  x
3)
x 2  49
5)
7 x  343
2)
25  x
4)
x3  125
6)
5x  3125
What is a log and when do I use it?
 A log is an ________________________.
 Use logs to solve for missing ________________________.
 Example: 10x = 150
 To find the exact value for x, you must convert this expression into a log.
Logs and Exponentials are Inverses of each other.
 exponential function: y = bx
 inverse of an exponential function: x = by
y is the logarithm of x:
y = logbx
“y = log base b of x”
Logarithmic Form:
:Exponential Form
**When the base of the exponent is 10, the subscript is not written, they are
Ex:
**When the base of the exponent is e, the logarithm is called
Ex:
and written
Examples:
Write the following in exponential form.
Write the following in logarithmic form.
1. log7 2401 = 4
1. 27 = 128
3
2. log81 27 = 4
2. 104 = 10,000
3. ln 10 ≈ 2.302
3. 7-2 = 49
4. ln 20.085 ≈ 3
4. e4 ≈ 54.598
5. log 5
1
125
1
= -3
5. 50 = 1
Properties of Logarithms
1. 𝑙𝑜𝑔𝑏 1 = 0 because 𝑏 0 = 1
Properties of Natural Logarithms
1. ln 1 = 0 because 𝑒 0 = 1
2. 𝑙𝑜𝑔𝑏 𝑏 = 1 because 𝑏1 = 𝑏
2. ln e = 1 because 𝑒 1 = 𝑒
3. 𝑙𝑜𝑔𝑏 𝑏 𝑥 = 𝑥 because 𝑏 𝑥 = 𝑏 𝑥
3. ln 𝑒 𝑥 = x because 𝑒 𝑥 = 𝑥
𝑏 𝑙𝑜𝑔𝑏𝑥 = 𝑥
𝑒 𝑙𝑛𝑥 = 𝑥
 Inverse Properties 
4. If 𝑙𝑜𝑔𝑏 𝑥 = 𝑙𝑜𝑔𝑎𝑏 𝑦 𝑡ℎ𝑒𝑛 𝑥 = 𝑦
 1 to 1 Property 
4. If ln x = ln y then x = y
Evaluating Log Expressions WITHOUT a calculator:
 Think: b to what power = the given number
 Set = to x and convert back to an exponential and solve
1
1) log 3 27
2) log 2 32
3) log 5
25
5)
log 9 3
6)
9)
log 8 8
10) log335
ln 1
7)
log 7 1
11) 4log4 7
4)
log1/ 2 8
8)
log 6 6
12) ln e-2
Evaluate with a calculator:
The
on the calculator will do base 10 logs (common logs).
1
2. log ≈
1. log 13≈
3. log -22 ≈
3
The
on the calculator will do base e log (natural logs).
3
2. ln ≈
1. ln 16≈
4
Practice Problems
Rewrite the following in exponential form.
1. 𝑙𝑜𝑔4 1024 = 5
3
5. 𝑙𝑜𝑔16 8 = 4
2. 𝑙𝑜𝑔5 5 = −1
1
3. 𝑙𝑜𝑔36 6 = − 2
1
1
6. ln 15 ≈ 2.71
7. ln 5 ≈ −1.61
1
4. 𝑙𝑜𝑔8 512 = 3
8. log 22.5 ≈ 1.35
Rewrite the following in logarithmic form.
3
1
9. 102 = 100
10. 92 = 27
11. 4−3 = 64
12. 53 = 125
13. 𝑒 7 ≈ 1096.63
14. 𝑒 −2 ≈ 0.14
15. 70 = 1
16. 61 = 6
Evaluate WITHOUT a calculator.
17. 𝑙𝑜𝑔7 49
18. 𝑙𝑜𝑔8 1
19. 𝑙𝑜𝑔12 12
20. 𝑙𝑜𝑔16 4
21. 𝑙𝑜𝑔3 81
22. 𝑙𝑜𝑔2 16
23. log 10
24. 𝑙𝑜𝑔1 8
26. 𝑙𝑜𝑔5 54
27. ln 𝑒 9
28. ln 𝑒 −4
1
25. 𝑙𝑜𝑔6 36
2