y = bx ≡ x = logb y
So, what is a logarithm? Google defines it as “a quantity representing the power to which a fixed number (the base) must be raised to produce a given number.”
101=10
102=100
103=1000
104=10000
A Problem We Are Trying To Solve
If we had 10x = 5000. What is the value of x?
Note: On my Casio FX-570W this is log(1) = 0
Log10 1 = 0
Log10 2 = 0.3103
Log10 3 = 0.4771
Log10 4 = 0.6021
Log10 5 = 0.6989
Log10 10 = 1
Log10 100 = 2
Log10 1000 = 3
We can infer from this that Log10 10k = k
The following are equivalence (≡) equations. I.e. the same but a different way of writing.
Log10 10 ≡ 101
Log10 100 ≡ 102
Log10 1000 ≡ 103
Exponential Form
y = bx
Logarithmic Form
x = logb y
Examples
5000 = 10x (form of y = bx ) ≡ x = Log10 5000 (form of x = logb y) ≡ 3.69897004…
Covert Exponential Form to Logarithmic Form
- 25 = 32
- 3-1 = 1/3
- 4096 = 46
My first step would be to convert the Exponential Forms to standard y = bx
- 25 = 32 ≡ 32 = 25
- 3-1 = 1/3 ≡ 0.3 = 3-1
- 4096 = 46 ≡ 46 = 4096
Therefore,
- 25 = 32 ≡ 32 = 25 ≡ 5 = Log2 32
- 3-1 = 1/3 ≡ 0.3 = 3-1≡ -1 = Log3 0.3
- 4096 = 46 ≡ 46 = 4096 ≡ 6 = Log4 4096
Covert Logarithmic Form to Exponential Form
- Log9 81 = 2
- Log8 4 = 2/3
- -1.5 = Log25 1/125
convert the Logarithmic Forms to standard x = Logb y
- Log9 81 = 2 ≡ 2 = log9 81
- Log8 4 = 2/3 ≡ 2/3 = Log8 4
- -1.5 = Log25 1/125 (already in standard form)
Therefore,
- Log9 81 = 2 ≡ 2 = log9 81 ≡ 81 = 92
- Log8 4 = 2/3 ≡ 2/3 = Log8 4 ≡ 4 = 82/3
- -1.5 = Log25 1/125 (already in standard form) ≡ 1/125 = 25-1.5