Logarithm product rule
log_b(x * y) = log_b(x) + log_b(y)

Logarithm quotient rule
log_b(x / y) = log_b(x) - log_b(y)

Logarithm power rule
log_b(x^y) = y * log_b(x)

The log is the inverse of the exponential function so 4³ =
is the inverse of log₄ 64 =

Product Rule of Logs

log₂ 4 + log₂ 8 = 5

We can also solve this by multiplying the bases together and get log₂ 32 = 5

When we add together logs that have the same bases we can combine them by multiplying the numbers we are taking the log of.

log_b (x * y) = log_b x + log_b y

log₆ 12 + log₆ 18 = log₆ 216 = 3

log₂ 32 = log₂ 4 + log₂ 8 = 2 + 3 + 5

---

The Power Rule of Logs

If we have log₇ x³ =

log₇ (x * x * x) =

log₇ x + log₇ (x * x) =

log₇ x + log₇ x + log₇ x =

3 (log₇ x) - so when we take the log of something that has an exponent we can move the exponent as a multiplier to the front of the expression.

log₂ 64 =
log₂ (4 * 4 * 4)
log₂ (4³) = 3 (log₂ 4) = 3 * 2 = 6

log_bxⁿ = n log_b x

- - - - - - - - - - - - - - - - - - - - - -
4^2x = 6

log (4^2x) = log(6)
2x log(4) = log(6)
2x = (log(6)/log(4)
x = log(6)/log(4)/2

- - - - - - - - - - - - - - - - - - - - - -

2^x+3 = 54

log (2^x+3) = log(54)
(x+3) log(2) = log(54)
x log(2) + 3 log(2) = log(54)
x log(2) = log(54) - 3 log(2) x = (log(54) - 3 log(2)) / log(2)
x = 2.7548

---

Exponentail Rule (Inverse of the log)

log₄ (x -1 ) = 3
4^log₄(x-1) = 4³
x-1 = 64
x = 65


log(x+2) = 4
10^log(x+2) = 10⁴
x + 2 = 10000
x = 9998

---

Logarithm Quotient Rule

log(30x) - log(6) = 4
log(30x/6) = 4
log(5x) = 4
10^log(5x) = 10⁴ // exponental rule
5x = 10000
x = 2000

- - - - - - - - - - - - - - - - - - - - - -


log₅ 25 - log₅ 5 =
log₅ (25/5) =
log₅ (5) = 1

---

---