# Exponential Growth (Part 2)

How to solve exponential growth problems using logarithmics: formulas, examples, and their solutions.

## Formula

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

Exponential growth (part 1)

By using the logarithm,

now you can find the value of *t*.

## Example 1

Write the given values to use.

Put the given values into the formula.

1000⋅(1 + 0.06)^{t} = 1800

Log both sides. (base: 1.06)

Logarithmic form

To use the given conditions,

change the base to 10.

Change the base formula

log 1.8 = 0.255, log 1.06 = 0.025

(given conditions)

*t* = 10.2 years

But the question says 'how many years'.

So *t* is a natural number.

So round 10.2 up to the nearest ones:*t* = 11 years.

This means after 11 years,

the investment worth more than $1,800.

## Formula: Continuous Compounding

Recall the continuous compounding formula:*A* = *A*_{0}*e*^{rt}

Exponential growth (part 1)

By using the logarithm,

you can also find *t* from this formula.

## Example 2

Write the given values to use.

Put the given values into the formula.

1000⋅*e*^{0.06⋅t} = 1800

Log both sides. (base: *e*)

Natural logarithms

ln 1.8 = 0.588

(given condition)

*t* = 9.8 years

But the question says 'how many years'.

So *t* is a natural number.

So round 9.8 up to the nearest ones:*t* = 10 years.

This means after 10 years,

the investment worth more than $1,800.

As you can see,

the answer is greater than

the previous example's answer:

the investment is maximized

by **continuous** compounding.