# Logarithms » Logarithm rules

You know that 2`log`(8) = 3, because 23 = 8.
This gives: 22`log`(8) = 8.

You know that 5`log`(78 125) = 7, because 57 = 78 125.
This gives: 55`log`(78 125) = 78 125.

g`log`(x) = y means gy = x.
Substituting y = g`log`(x) into gy = x, gives gg`log`(x) = x.

For g > 0, g ≠ 1, a > 0 and b > 0 the following rules are true:

g`log`(a) + g`log`(b) = g`log`(ab)
g`log`(a) – g`log`(b) = g`log`(ab)

n · g`log`(a) = g`log`(an)

g`log`(a) = p`log`(a)p`log`(g) = `log`(a)`log`(g)

1g`log`(a) = – g`log`(a)

Example 1
Reduce 5 – 3 · 2`log`(3) to one logarithm.

5 – 3 · 2`log`(3) =
2`log`(25) – 2`log`(33) =
2`log`(2533) =
2`log`(3227)

Example 2
Solve the equation  1 + 2 · 5`log`(x) = 7  algebraically.

 1 + 2 · 5`log`(x) = 7 2 · 5`log`(x) = 6 5`log`(x) = 3 x = 53 x = 125
Solve the equation 2 · 2`log`(x) + 0,5`log`(x + 6) = 0 algebraically.
 2 · 2`log`(x) + 0,5`log`(x + 6) = 0 2`log`(x2) + 2`log`(x + 6)2`log`(0,5) = 0 2`log`(x2) – 2`log`(x + 6) = 0 2`log`(x2) = 2`log`(x + 6) x2 = x + 6 x2 – x – 6 = 0 (x + 2)(x – 3) = 0 x = –2 of x = 3