ASTB- Formulas and Other Info

Log product rule

log_b(xy) = log_b(x) + log_b(y)

Log quotient rule

log_b(x/y) = log_b(x) - log_b(y)

Log power rule

log_b(x^n) = n log_b(x)

Log of 1

log_b(1) = 0

Log of its own base

log_b(b) = 1

Log and exponent cancel

log_b(b^x) = x

Change-of-base rule

log_b(x) = log(x)/log(b)

Adding two logs with the same base

Multiply their arguments: log_b(x) + log_b(y) = log_b(xy)

Subtracting two logs with the same base

Divide their arguments: log_b(x) - log_b(y) = log_b(x/y)

Can logs with different bases combine directly?

No, not using the product or quotient rules

What does log(x) = 3 mean?

x = 10^3 = 1000 (common log)

Fractional exponent rule

a^(m/n) = (nth root of a)^m

Negative exponent rule

a^(-n) = 1/a^n, for a not equal to 0

Square root simplification rule

Extract perfect-square factors: sqrt(12) = 2sqrt(3)

Rectangle perimeter

P = 2l + 2w

Rectangle area

A = lw

Square perimeter

P = 4s

Square area

A = s^2

Cube volume

V = s^3

Cube surface area

SA = 6s^2

Rectangular prism volume

V = lwh

Rectangular prism surface area

SA = 2(lw + lh + wh)

Triangle area

A = 1/2 bh

Triangle interior angles

Sum = 180 degrees

Pythagorean theorem

a^2 + b^2 = c^2; c is the hypotenuse

Circle circumference

C = 2 pi r = pi d

Circle area

A = pi r^2

Circle diameter

d = 2r

Arc length in radians

s = r theta

Arc length in degrees

s = (theta/360) x 2 pi r

Sector area in radians

A = 1/2 r^2 theta

Sector area in degrees

A = (theta/360) x pi r^2

Degrees to radians

Radians = degrees x pi/180

Radians to degrees

Degrees = radians x 180/pi

Cylinder volume

V = pi r^2 h

Triangular prism volume

V = (1/2 bh) x prism length

Pyramid or cone volume

V = 1/3 Bh

Sphere volume

V = 4/3 pi r^3

Distance formula

d = vt

Acceleration formula

a = (final velocity - initial velocity)/time

Newton's first law

Law of inertia: motion stays unchanged unless a net external force acts

Newton's second law

Law of acceleration: F = ma

Newton's third law

Action-reaction: equal and opposite forces act on different objects

Force

F = ma; unit is the newton (N)

Weight near Earth's surface

W = mg; g is approximately 9.8 m/s^2

Momentum

p = mv

Work with parallel force

W = Fd

Work with angled force

W = Fd cos(theta)

Power

P = W/t

Power when force and velocity align

P = Fv

Kinetic energy

KE = 1/2 mv^2

Gravitational potential energy

PE = mgh

Conservation of mechanical energy

KE + PE stays constant when only conservative forces act

Pressure

P = F/A

Density

rho = m/V

Torque for perpendicular force

tau = rF

Efficiency

Efficiency = useful output energy / input energy x 100%

Mechanical advantage

MA = output force / input force

Lever balance rule

Effort force x effort arm = load force x load arm

Ideal simple machine

Input work = output work

Ohm's law

V = IR

Electric power

P = VI; also P = I^2 R or V^2/R

Series circuit current

Same current through each component; a break stops the loop

Parallel circuit voltage

Same voltage across each branch

Heat energy

Q = mc delta T

Conduction

Heat transfer by direct contact

Convection

Heat transfer by movement of fluids

Radiation

Heat transfer by electromagnetic waves; no medium needed

Melting

Solid to liquid

Freezing

Liquid to solid

Vaporization

Liquid to gas

Condensation

Gas to liquid

Sublimation

Solid directly to gas

Deposition

Gas directly to solid

Ionization

Gas to plasma

Recombination (deionization)

Plasma to gas

Endothermic phase changes

Absorb heat: melting, vaporization, sublimation

Exothermic phase changes

Release heat: freezing, condensation, deposition

Absolute zero

0 K = -273.15 degrees C

Celsius to Kelvin

K = degrees C + 273.15

Conservation of energy

Energy cannot be created or destroyed, only transferred or transformed

Conservation of momentum

Total momentum remains constant in an isolated system