# Exponential Growth (Part 2)

How to find the amount of time for the exponential growth (compounding yearly, continuously) by using logarithm: formulas, examples, and their solutions.

## Formula

Recall the exponential growth formula:*A* = *A*_{0}(1 + *r*)^{t}.

Exponential Growth - Final Value

By using the logarithm,

now you can find the value of *t*.

## Example 1

The amount of initial investment is $1,000.

So *A*_{0} = 1000.

The amount of final investment is $1,800.

So *A* = 1800.

The rate is 6% per year,

compounded [yearly].

So *r* = +0.06/year.

*A*_{0} = 1000*A* = 1800*r* = +0.06/year

Then 1000⋅(1 + 0.06)^{5} = 1800.

Divide both sides by 1000.

Then 1.06^{t} = 1.8.

Log both sides.

(base: 1.06)

Then *t* = log_{1.06} 1.8.

Logarithmic form

log_{1.06} 1.8 = (log 1.8)/(log 1.06)

Change the base formula

log 1.8 = 0.255

log 1.06 = 0.025

Then (given) = 0.255/0.025.

Move the decimal points of both numbers

3 digits to the right.

(= Multiply 1000

to both of the numerator and the denominator.)

255/25 = 10.2

So *t* = 10.2.

*t* = 10.2 years

But *t* should be a natural number.

So round 10.2 up to the nearest ones:

10.2 → 11.

So *t* = 11 years.

So the answer is [after 11 years].

## Formula: Continuous Compounding

Recall the continuous exponential growth formula:*A* = *A*_{0}*e*^{rt}.

Continuous Exponential Growth - Final Value

By using the logarithm,

you can also find *t* from this formula.

## Example 2

The amount of initial investment is $1,000.

So *A*_{0} = 1000.

The amount of final investment is $1,800.

So *A* = 1800.

The rate is 6% per year,

compounded [yearly].

So *r* = +0.06/year.

*A*_{0} = 1000*A* = 1800*r* = +0.06/year

Then 1000⋅*e*^{0.06⋅t} = 1800.

Divide both sides by 1000.

Then *e*^{0.06t} = 1.8.

Log both sides.

(base: *e*)

Then 0.06*t* = ln 1.8.

Natural logarithm

ln 1.8 = 0.588

Divide both sides by 0.06.

Then *t* = 0.588/0.06.

Move the decimal points of both numbers

2 digits to the right.

(= Multiply 100

to both of the numerator and the denominator.)

58.8/6 = 9.8

So *t* = 9.8.

*t* = 9.8 years

But *t* should be a natural number.

So round 9.8 up to the nearest ones:

9.8 → 10.

So *t* = 10 years.

So the answer is [after 10 years].

Compare the answer 9.8 years (continuously)

to the answers of the last example:

10.2 years (yearly).

As you can see,

by compounding the investment continuously,

the amount of time to reach $1,800 decreases.

This is true because

by compounding continuously,

the investment grows faster.