Exponents, Logs, Equations of them

Facts of exponents:
b^u+v ↔ b^u·b^v      Exs:    2³⁺⁴ = 2³·2⁴    x³⁺⁴ = x³·x⁴    2^x+y = 2^x·2^y
b^u-v ↔ b^u / b^v      Exs:    2^7-4 = 2⁷ / 2⁴    x^7-4 = x⁷ / x⁴    2^x-y = 2^x / 2^y
(b^u)^v ↔ b^uv      Exs:    (2³)⁴ = 2^3·4    (x³)⁴ = x^3·4    (2^x)^y = 2^x·y   

Facts of logarithms: for every base
log uv ↔ log u + log v      Exs:    log 3x = log 3 + log x
log u/v ↔ log u - logv      Exs:    log 3/x = log 3 - log x
log u^v ↔ v·log u      Exs:    log 3^x = x log 3    log x³ = 3 log x
→ expand, split
← reassemble, combine, compactify

Equations.

If both sides are exponential and the bases are the same or can be rewritten as the same base:
Use: b^lhs = b^rhs → lhs = rhs
Ex. 4^x = 2³    Rewrite: (2²)^x=2^2x = 2³, 2x=3, x=3/2

Equations:
lhs = rhs ↔ b^lhs = b^rhs      Raise both sides of equation as exponents of some base b or set the exponents as equal.
lhs = rhs ↔ log_blhs = log_brhs      Take logs of both sides of equation or set the logs' arguments as equal.

Solve equation for variable x:
If x is in exponent in equation i.e. b^x, take log_b of each side of the equation to get the x out of exponent (the log will "undo" the exponent):
log_bb^x → = x (the exponent to raise b to to get b^x is x),
Inverse functions: log(exp(x))=x
Ex.
5^x = 100
log₅ 5^x = log₅ 100
x = log₅ 100
x ≈ 2.86135
Ex.
e^2x = 100
ln e^2x = ln 100
2x = ln 100
x = (ln 100) / 2
x ≈ 2.30258

If x is in logarithm in equation i.e. log_bx, raise the base b to both sides of the equation i.e. b^log_bx to get the x out of log (the exponentiation will "undo" the logarithm):
b^log_bx → = x (raise b to the exponent that b has to be raised to to get x is x)
Inverse functions: exp(log(x))=x
Ex.
log₅2x = 3
5^log₅2x = 5³
2x = 125
x = 62.5
Ex.
ln 2x = 3
e^{ln 2x} = e³
2x = e³
x ≈ 10.04276