precal

Multiplying powers with the same base

Add exponents: x^m · x^n = x^(m+n)

Dividing powers with the same base

Subtract exponents: x^m / x^n = x^(m−n)

Power raised to a power

Multiply exponents: (x^m)^n = x^(mn)

Power of a product

Apply the exponent to every factor: (xy)^n = x^n · y^n

Power of a fraction

Apply the exponent to the top and bottom: (x/y)^n = x^n / y^n

Negative exponent

Take the reciprocal: x^(−n) = 1/x^n

Zero exponent

Any nonzero number raised to zero equals 1: x^0 = 1

Fractional exponent

Bottom is the root and top is the power: a^(m/n) = (ⁿ√a)^m

Logarithm meaning

log_b(A) = x means b^x = A

Log of 1

log_b(1) = 0

Log of its own base

log_b(b) = 1

Natural logarithm

ln is a logarithm with base e

Natural log cancels e

ln(e^x) = x

Log product rule

log_b(MN) = log_b(M) + log_b(N)

Log quotient rule

log_b(M/N) = log_b(M) − log_b(N)

Log power rule

log_b(M^p) = p · log_b(M)

Expanding logs

Split division into subtraction → split multiplication into addition → move powers in front

Condensing logs

Move coefficients into powers → multiply added terms → divide by subtracted terms

Expanding example

log(x³y²/z⁴) = 3log(x) + 2log(y) − 4log(z)

Condensing example

2ln(x) + 3ln(y) − ln(z) = ln(x²y³/z)

Log of a root

ln(ⁿ√x) = (1/n)ln(x)

Addition inside a log

log(x + y) cannot be split into log(x) + log(y)

Combining logs

Logs must have the same base

Log domain

The expression inside each log must be positive

Solving e^(kx) = C

x = ln(C)/k

Solving b^(kx) = C

x = ln(C)/(k · ln(b))

Before taking ln

Isolate the exponential by subtracting constants and dividing by its coefficient

Rewriting ln(A)/n

ln(A)/n = ln(A^(1/n))

Compound interest formula

A = P(1 + r/n)^(nt)

P in compound interest

Starting amount of money

A in compound interest

Final amount of money

r in compound interest

Annual interest rate written as a decimal

n in compound interest

Number of times interest compounds per year

t in compound interest

Time in years

Percent to decimal

Divide by 100: 1.25% = 0.0125

Compounding annually

n = 1

Compounding quarterly

n = 4

Compounding monthly

n = 12

Compounding daily

n = 365

Exponential model

y = a · b^t

a in an exponential model

Initial value when t = 0

b in an exponential model

Multiplier for each one-unit increase in time

Exponential growth formula

y = a(1 + r)^t

Exponential decay formula

y = a(1 − r)^t

r in growth and decay formulas

Rate written as a decimal

Growth rate from multiplier b

r = b − 1 when b > 1

Decay rate from multiplier b

r = 1 − b when 0 < b < 1

Finding a decay rate from two amounts

r = 1 − (final/initial)^(1/t)

Decimal to percent

Multiply by 100: 0.18 = 18%

Half-life formula

N = N₀(1/2)^(t/h)

N in half-life

Amount remaining

N₀ in half-life

Original amount

h in half-life

Time required for half the amount to remain

75% remains

N = 0.75N₀

Solving for time with half-life

t = h · ln(N/N₀)/ln(1/2)

Standard logistic model

P(t) = L/(1 + B · e^(−kt))

Initial value of a standard logistic model

P(0) = L/(1 + B)

Initial value with a different logistic denominator

For P(t) = A/(c + d · e^(−kt)) use P(0) = A/(c + d)

Finding time in a logistic model

Set P(t) equal to the target → clear the denominator → isolate the exponential → take ln → solve

Initially

Substitute t = 0

e to the zero power

e^0 = 1

Y-intercept of y = a · b^x

At x = 0 the value of y is a

Another easy exponential graph point

At x = 1 the value of y is a · b

Horizontal asymptote of y = a · b^x

y = 0

Positive exponential growth graph

When a > 0 and b > 1 the graph rises to the right

Positive exponential decay graph

When a > 0 and 0 < b < 1 the graph falls to the right

Negative a in an exponential graph

Reflects the graph across the x-axis

Finding b from two points

b = (y₂/y₁)^(1/(t₂ − t₁))

Finding a after finding b

a = y₁/(b^t₁)

Finding b when time values are two units apart

b = √(y₂/y₁)

Rounding answers

Keep full calculator precision until the final step