Logarithms are inverse operations of exponentiation.

For example, since 2⁴ = 16,

log₂16 = 4

Similarly, log₂ of 8 is 3.

Raising a number to a positive power increases the number, whereas a logarithm of a number decreases the number.

Properties

If we take the log of an exponential, and the bases are the same, then the output is the exponent. More precisely:

log_b(b^p) = p

For example:

log₂(2³) = 3

We can confirm this by evaluating the operation on the left. We know that:

2³ = 8

We also know that:

log₂8 = 3

Thus,

log₂(2³) = 3

Here are some more examples:

b^(log_bx) = x

2^(log₂8) = 8

2³ = 8

property 2:

log_b(x ⋅ y) = (log_bx) + (log_by)

log₂(8 ⋅ 16) = (log₂8) + (log₂16)

log₂(8 ⋅ 16) = 3 + 4

log₂(8 ⋅ 16) = 7

This was useful when it was hard to compute x*y. We can take the log of both sides with base 10. Then we can compute the right side by consulting a log table, then adding the logs. Then we can use a "anti-log" table, which tells us how to compute 10^x.

For example, suppose we want to calculate 14*18.

Then first take the log of it:

log₁₀(14 ⋅ 18)

This is equal to

(log₁₀14) + (log₁₀18)

Now we can look up the log table. It tells us that

log₁₀14 = 1.146

log₁₀18 = 1.255

Then we add them:

2.401

Finally, we look up the anti-log table which tells us:

10^2.401 = 251.77

14*18 is actually 252, so this was quite close.

Property 3

log_b(x^p) = p ⋅ (log_bx)

log₂(8³) = 3 ⋅ (log₂8) == 3*3 == 9

log₂512 = 9

Log of Power property:

log_b(b^p) = p

log₂(2³) = 3

Changing the Base:

log_bx = (log_kx) / (log_kb)

log₂8 = (log₁₀8) / (log₁₀2)