Home > Indices and logs, Maths > Logs, the exponential function and continuous growth

## Logs, the exponential function and continuous growth

You will probably have come across the principle of continuous compounding at A level or GCSE.

Say you have £1000 to invest in a savings account that offers an interest rate of 3%. What is your account worth after 10 years?

If the interest is compounded annually, then its $1000(1.03)^{10}=1343.92$

If the interest is compounded monthly, then its $1000(1.0025)^{120}=1349.35$

If the interest is compounded weekly, then its $1000(1.000577)^{520}=1349.74$

The formula you are using here is $P_t = P_0(1+\frac {r}{n})^{nt}$ where r is the annual rate of interest expressed as a decimal (so 3% is 0.03), t is the number of years and n is the frequency of compounding

As n tends to infinity this tends to the continuous compounding formula, $P_t = P_0 e^{rt}$

So if the interest is compounded continuously, then its $1000e^(0.3)=1349.86$

When you have questions related to the size of an economy (its GDP) then its growth rate will be a continuous growth rate, the economy doesn’t grow in big chunks at the end of the year, or a month etc, it is growing all the time.

So if you say the US economy is approximately twice the size of the Chinese economy in 2011, and assume the US will grow at a constant rate of 3% and China at a constant rate of 9%, when will the Chinese economy be bigger than the US economy?

Well you can set the problem out in three equations:
$USGDP_{2011} = 2ChinaGDP_{2011}$ (1)

$USGDP_t = USGDP_{2011}e^{0.03t}$ (2)

$ChinaGDP_t = ChinaGDP_{2011}e^{0.09t}$ (3)

At the point when China has the same GDP as the US, $USGDP_{2011}e^{0.03t} = ChinaGDP_{2011}e^{0.09t}$

Equation (1) allows us to express this purely in terms of China GDP: $2ChinaGDP_{2011}e^{0.03t} = ChinaGDP_{2011}e^{0.09t}$

Divide both sides by the value of China’s GDP in 2011: $2e^{0.03t} = e^{0.09t}$

Now take logs of both sides: $ln (2) + 0.03t (ln (e)) = 0.09t (ln (e))$

ln e = 1, so this becomes $ln (2) + 0.03t = 0.09t \Rightarrow ln (2) = 0.06t \Rightarrow t = \frac{ln (2)}{0.06} = 11.552$

So this tells us that when t (the number of years) is 11.552, China would catch up with the USA (and be about to overtake it). As our starting year was 2011, this will mean that some point in the middle of 2022 China would overtake the USA.