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Time to reach a level

The question of when an exponential model reaches a given value is an equation with $x$ in the exponent. You divide out the initial value and take the logarithm. The answer is rarely a whole number of periods. If the amount must have passed the level, you round up to the next whole period. Half-life and doubling time are special cases where the ratio of final value to initial value is $1/2$ or $2$ .

a\cdot b^x = c \Rightarrow x = \frac{\ln(c/a)}{\ln b}

the time

x = \frac{\ln 2}{\ln b}

doubling time

Symbols

$x$	number of periods
$c$	the level you want to reach
$b$	growth factor

Example

1000 kr at 8% interest doubles when $1.08^x = 2$ . Then $x = \dfrac{\ln 2}{\ln 1.08} \approx 9.01$ years.

Use the same logarithm in the numerator and the denominator. The ratio is the same for lg and ln.

Practise exponential models for free →

← Percent and percentage points

Part of Mathematics 1T: Exponential models.