Logarithm: Log base b of a number x is the exponent to which b must be raised to get x.
Notation: log_b(x) = y means b^y = x.
Examples of Logarithms
log_2(16):
2^4 = 16
Result: log_2(16) = 4
log_3(27):
3^3 = 27
Result: log_3(27) = 3
log_5(25):
5^2 = 25
Result: log_5(25) = 2
log_4(1):
Always equals 0
Result: log_4(1) = 0
log_7(7):
7^1 = 7
Result: log_7(7) = 1
Logarithm of Powers of 10
log(1000) (base 10 assumed):
10^3 = 1000
Result: log(1000) = 3
log(100):
10^2 = 100
Result: log(100) = 2
log(0.0001):
10^-4 = 0.0001
Result: log(0.0001) = -4
Patterns in Logarithms
log(1) = 0
log(10) = 1
log(100) = 2
log(1000) = 3
Pattern: For each additional zero, the logarithm increases by 1.
For negative decimals:
log(0.1) = -1
log(0.01) = -2
log(0.001) = -3
log(0.0001) = -4
More Logarithmic Identities
log_b(x) = y implies b^y = x
Example: 7 log_7(38) = 38
Example: 5 log_5(14) = 14
Example: 8 log_8(y) = y
Additional Examples
log_3(9):
3^2 = 9
Result: log_3(9) = 2
log_3(1/9):
3^-2 = 1/9
Result: log_3(1/9) = -2
log_9(3):
Result: log_9(3) = 1/2
log_9(1/3):
Result: log_9(1/3) = -1/2
Practice Problems
2^x = 8:
x = 3
3^(-2) = 1/9:
Result: Log value = -2
Square root of 25:
25^(1/2) = 5
Result: 5
Square root of 64:
64^(1/2) = 8
Result: 8
4th root of 16:
16^(1/4) = 2
Result: 2
4th root of 81:
81^(1/4) = 3
Result: 3
Conclusion
Logarithms are an essential concept in mathematics that relate exponentiation and multiplication. Understanding their properties and how to evaluate them is crucial.