Trying to figure out my percentage raises at work

Question

If I make $ 14.40 right now and max out at 24 an hour, how many seperate 2% raises will I need to get to be at the max? Is there a simple math equation to get this answer or do I have to just keep multiplying 2% to each new wage and then add up how many times it took?

Answer 1

You are looking for a solution to the equation

$$14.4\cdot 1.02^x = 24$$

and here $\log_{1.02}$ yields the solution

$$x = log_{1.02} \left( \frac {24}{14.4} \right) \approx 25.796$$

edit: There was an error in my previous result, that has been corrected.

Answer 2

You want to solve the following equation for $n$:

$$14.40\times 1.02^n = 24.00$$ $$\ln(14.4\times1.02^n) = \ln(24)$$ Apply the rules to separate products and bring down the exponent $$\ln(14.4) + n\ln(1.02) = \ln(24)$$ Subtract from both sides, then combine the logarithms $$n\ln(1.02) = \ln(24) - \ln(14.4) = \ln\left(\frac{24}{14.4}\right)$$ Then divide by the remaining coefficient to isolate $n$: $$n = \frac{\ln\left(\frac{24}{14.4}\right)}{\ln(1.02)} \approx 25.79$$

Which means it will take you $25$ raises to bring you just under the limit, and $26$ to go over.

Answer 3

If the wage of $\$ 14.4$ raises one time, the new wage is $\$ 14.4 \cdot 1.02$

If the wage of $\$ 14.4$ raises two times, the new wage is $\$ 14.4 \cdot 1.02^2 $

If the wage of $\$ 14.4$ raises n times, the new wage is $\$ 14.4 \cdot 1.02^n $

Thus your equation is $\$ 14.4 \cdot 1.02^n=\$ 24$

Dividing both sides of the equation by $\$ 14.4$

$1.02^n=\frac{24}{14.4}$

Taking logs

$n\cdot \log(1.02)=\log\left( \frac{24}{14.4} \right)$

${\large{n=\frac{ \log\left( \frac{24}{14.4} \right)}{{\log(1.02)}}}}\approx 25.8$

Since you want to have at least $\$ 24$, you need 26 raises.