Answer:
[tex]\displaystyle f(x) = \ln\left( 1 + x^2\right)[/tex]
Step-by-step explanation:
We are given that:
[tex]\displaystyle e^{f(x)} = 1 + x^2[/tex]
And we want to find f(x).
We can take the natural log of both sides:
[tex]\displaystyle \ln\left(e^{f(x)}\right) = \ln\left( 1 + x^2\right)[/tex]
Recall that ln(aᵇ) = bln(a). Hence:
[tex]\displaystyle f(x) \ln(e) = \ln\left ( 1 + x^2\right)[/tex]
By definition ln(e) = 1. Therefore:
[tex]\displaystyle f(x) = \ln\left( 1 + x^2\right)[/tex]
