Logarithms – Exponential Form and Logarithmic Form

y = b^x  ≡ x = log_b 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 10^x = 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 10^k = 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 = b^x

Logarithmic Form

x = log_b y

Examples

5000 = 10x  (form of y = b^x ) ≡ x = Log₁₀ 5000 (form of x = log_b 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 = b^x

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 ≡ 5 = 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 = Log_b 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 = 8^2/3
3. -1.5 = Log₂₅ 1/125 (already in standard form) ≡ 1/125 = 25^-1.5