## 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 ≡ 10
Log10 100 ≡ 102
Log10 1000 ≡ 10

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

1. 25 = 32
2. 3-1 = 1/3
3. 4096 = 46

My first step would be to convert the Exponential Forms to standard y = bx

1. 25 = 32 ≡ 32 = 25
2. 3-1 = 1/3 ≡ 0.3 = 3-1
3. 4096 = 46 46 = 4096

Therefore,

1. 25 = 32 ≡ 32 = 25 = Log2 32
2. 3-1 = 1/3 ≡ 0.3 = 3-1-1 = Log3 0.3
3. 4096 = 46 ≡ 46 = 4096 ≡ 6 = Log4 4096

Covert Logarithmic Form to Exponential Form

1. Log9 81 = 2
2. Log8 4 = 2/3
3. -1.5 = Log25 1/125

convert the Logarithmic Forms to standard x = Logb y

1. Log9 81 = 2 ≡ 2 = log9 81
2. Log8 4 = 2/3 ≡ 2/3 = Log8 4
3. -1.5 = Log25 1/125 (already in standard form)

Therefore,

1. Log9 81 = 2 ≡ 2 = log9 81 ≡ 81 = 92
2. Log8 4 = 2/3 ≡ 2/3 = Log8 4 ≡ 4 = 82/3
3. -1.5 = Log25 1/125 (already in standard form) ≡ 1/125 = 25-1.5