Question

Takumi plants a tree in his backyard and studies how the number of branches grows over time.

He predicts that the relationship between N, the number of branches on the tree, and t, the elapsed time, in years, since the tree was planted can be modeled by the following equation.
N=5*10^0.3t
According to Takumi's model, in how many years will the tree have 100 branches?
Give an exact answer expressed as a base-10 logarithm.

Answer 1

`(10)/(3)\log_(10)(20)\; \sf years`

Explanation:

To find out in how many years the tree will have 100 branches according to Takumi's model, substitute N = 100 into the given equation and solve for t.

`5\cdot 10^(0.3t)=100`

Divide both sides of the equation by 5:

`(5\cdot 10^(0.3t))/(5)=(100)/(5)`

`10^(0.3t)=20`

Take the logarithm with base 10 of both sides of the equation:

`\log_(10)\left(10^(0.3t)\right)=\log_(10)(20)`

Apply the Power Rule, logₐ(xⁿ) = n logₐ(x), to the left side of the equation:

`0.3t\log_(10)(10)=\log_(10)(20)`

Apply the logarithmic property logₐ(a) = 1:

`0.3t(1)=\log_(10)(20)`

`0.3t=\log_(10)(20)`

Rewrite 0.3 as 3/10:

`(3)/(10)t=\log_(10)(20)`

Multiply both sides of the equation by 10:

`3t=10\log_(10)(20)`

Divide both sides of the equation by 3:

`t=(10)/(3)\log_(10)(20)`

Therefore, the tree will have 100 branches in exactly ¹⁰/₃log₁₀(20) years.