Property 1

log_a1 = 0

      Recall that if a ≠0, a⁰ = 1 therefore, log_a1 = 0

      Hence, the logarithm of 1 to any base is 0.

Example

log₂1 = 0; because in the exponential equation

we know that 2⁰ = 1

log_1/21 = 0; because in the exponential equation

we know that (½)⁰ = 1

The logarithm of unit to any non–zero base is zero ( where unity = 1)

Property 2

log_aa = 1

      Recall that in exponents a1 = a ⇒log_aa =1

                                        (where a ≠0, a > 0)

Example

log₁₀10 = 1 because in the exponential equation

we know that 10¹ = 10

log7⁷ = 1 because in the exponential equation

we know that 7¹ = 7

log_ae = 1 because in the exponential equation

we know that e¹= e

The logarithm of any non–zero positive number to the same base is unity.

Property 3

log_aa^x = x

Recall that a^x = a^x

Example

Since you know that 3⁴ = 3⁴, you can write the logarithm equation as log₃3⁴ = 4

Try these questions

Find the following

1. log₃81
   Answer: log₃81 = log₃3⁴ = 4



2. log₆1
   Answer: log₆1 = x
        6^x = 1   ⇒ x = 0



3. log₁x = 23 Find x
   Answer: log₁x = 23   ⇒ 1²³ = 1
    ∴ x = 1


4. log₂256
   Answer: log₂256 = log₂2⁸ = 8


5. log₅25
   Answer: log₅25 = log₅5² = 2


6. If log_x8 = 3 then find x
   Answer: log_x8 = 3   ⇒ x³ = 8
                        x³ = 2³
                         x = 2