if the inflation rate is 4.5% compounded annually, how long will it take for prices to double?\n\nblank…

Explanation:

Step1: Set up the compound - interest formula

The compound - interest formula is $A = P(1 + r)^t$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate (as a decimal), and $t$ is the number of years. We want prices to double, so $A = 2P$, and $r=0.045$. Substituting these values into the formula gives $2P=P(1 + 0.045)^t$.

Step2: Simplify the equation

Divide both sides of the equation $2P = P(1.045)^t$ by $P$ (since $P\neq0$). We get $2=(1.045)^t$.

Step3: Take the natural logarithm of both sides

$\ln(2)=\ln(1.045^t)$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we can rewrite the right - hand side as $t\ln(1.045)$. So, $\ln(2)=t\ln(1.045)$.

Step4: Solve for $t$

$t=\frac{\ln(2)}{\ln(1.045)}$.

Step5: Calculate the value of $t$

We know that $\ln(2)\approx0.6931$ and $\ln(1.045)\approx0.044$. Then $t=\frac{0.6931}{0.044}\approx15.5$.

Answer:

$15.5$