$1500 is invested at 6% compounded continuously. How long until it… Step by Step

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The question

Question: $1500 is invested at 6% compounded continuously.

How long until it triples?

1 Write the equation

Continuous interest: A = P times e to the power rt.

A = Pe^rt

We know the start, $1500, and the end, $4500.

t is the unknown, up in the power.

1500 × e^0.06t = 4500

2 Get the power alone

To remove times 1500, we divide by 1500.

We do it on both sides.

Why both sides?

The = sign means both sides are the same amount.

If we change only one side, they are not the same any more.

~~1500~~ × e^0.06t~~1500~~ = 45001500

1500 divided by 1500 is 1.

So just the power is left.

4500 divided by 1500 is 3.

Now we have e to the power 0.06t = 3.

e^0.06t = 3

Careful!

When the power comes down, all of it comes down: 0.06t, not just t.

Careful: the whole power 0.06t comes down, not only t

3 Take ln of both sides

Logs bring a power down.

With e, we use ln.

So we take ln of both sides.

We do it on both sides, so they stay equal.

ln(e^0.06t) = ln 3

4 Bring the power down

ln and e undo each other: ln of e to the power k is just k.

ln(eᵏ) = k

So the left side is just 0.06t.

0.06t = ln 3

5 Get t alone

To remove times 0.06, we divide by 0.06.

We do it on both sides, so they stay equal.

t = ln 30.06

6 Use a calculator

With a calculator: ln 3 is about 1.09861.

ln 3 ≈ 1.09861

So t is about 18.31 years, to two decimal places.

t ≈ 18.31

7 Check

Let's check: put t = 18.31 back in.

With a calculator.

We get about $4499.94.

1500 × e^{0.06 × 18.31} ≈ 4499.94

That is 4500, up to the rounding.

It works!

≈ 4500

Let's look back.

First: write the equation.

Then: get the power alone.

Then: take ln of both sides.

Then: bring the power down.

Then: get t alone.

Then: use a calculator.

Last: check.

Answer