Answer:
[tex]r=\left(\dfrac{y}{P}\right)^{\frac{1}{x}}-1[/tex]
Step-by-step explanation:
Given exponential function:
[tex]f(x)=P(1+r)^x[/tex]
Replace f(x) with y:
[tex]y=P(1+r)^x[/tex]
Divide both sides by P:
[tex]\dfrac{y}{P}=(1+r)^x[/tex]
Take natural logs of both sides:
[tex]\ln \left(\dfrac{y}{P}\right)=\ln (1+r)^x[/tex]
[tex]\textsf{Apply the power law}: \quad \ln x^n=n \ln x[/tex]
[tex]\ln \left(\dfrac{y}{P}\right)=x \ln (1+r)[/tex]
Divide both sides by x:
[tex]\dfrac{1}{x}\ln \left(\dfrac{y}{P}\right)=\ln (1+r)[/tex]
[tex]\textsf{Apply the power law}: \quad n \ln x=\ln x^n[/tex]
[tex]\ln \left(\dfrac{y}{P}\right)^{\frac{1}{x}}=\ln (1+r)[/tex]
Raise both sides to power of e:
[tex]\large\text{$e^{\ln \left(\frac{y}{P}\right)^{\frac{1}{x}}}=e^{\ln (1+r)}$}[/tex]
[tex]\textsf{Apply the power law}: \quad e^{\ln x}=x[/tex]
[tex]\left(\dfrac{y}{P}\right)^{\frac{1}{x}}=1+r[/tex]
Subtract 1 from both sides:
[tex]\left(\dfrac{y}{P}\right)^{\frac{1}{x}}-1=r[/tex]
Therefore, the equation to find r is:
[tex]r=\left(\dfrac{y}{P}\right)^{\frac{1}{x}}-1[/tex]
