Math help please? Algebra 2?

A(t) = A₀e^(-kt)

A(t) = (1/2)A₀ when t = 5,730 [a half life is how long it takes for half of the original amount to be left]

=> (1/2)A₀ = A₀e^(-5,730k)

divide both sides by A₀

=> (1/2) = e^(-5,730k)

take the natrual og of both sides

=> ln(1/2) = ln(e^(-5,730k) )

given log_a(a^n) = n

=> ln(1/2) = -5,730k

=> ln(2^(-1)) = -5,730k

given log(a^n) = nlog(a)

=> -ln(2) = -5,730k

divide both sides by -5,730

=> k = ln(2)/5,730

=> k = ≈ 0.00012 (5 dp)

note A(t) = A₀e^(-tln(2)/5,730)

=> A(t) = A₀2^(-t/5,730)

Half life of as substance is defined as the time required to reduce the amount of original substance to half .

A(t) = A0 e^-kt

A(5730) = A0 e^-k(5730)

A0 /2 = A0 e^-k(5730)

1/2 = e^-k(5730)

ln(1/2) = -k(5730) ln(e)

k = -ln(1/2)/5730 = 0.00012

A(t)=A(0)e^-kt

A(5730)=A(0)/2=>

A(0)/2=A(0)e^(-5730k)=>

0.5=e^(-5730k)=>

-5730k=ln(0.5)=>

k=0.00012

1/2 = e^(-5730 k)

Take the natural log of both sides:

-0.693147 = -5730 k

Divide by -5730:

k = 0.00012097 = 1.2097 x 10^-4