Problems from Exponential functions

Development:

1. To find the base of the exponential function we raise and solve the following equation: $$x^2=16\Rightarrow x=4$$ Therefore $4$ is the base of the function, with $Dom(f)=\mathbb{R}$ and $Im(f)=(0,+\mathrm{\infty })$.

2. We proceed as in the previous case: $$x^{-2}=25\Rightarrow x^2={\displaystyle \frac{1}{25}}\Rightarrow x={\displaystyle \frac{1}{5}}$$ Therefore the base of the function is ${\displaystyle \frac{1}{5}}$, with $Dom(f)=\mathbb{R}$ and $Im(f)=(0,+\mathrm{\infty })$.

3. We proceed as in the previous case: $$x^3={\displaystyle \frac{1}{64}}\Rightarrow x=\sqrt[3]{{\displaystyle \frac{1}{64}}}\Rightarrow x={\displaystyle \frac{1}{4}}$$ Therefore the base of function is ${\displaystyle \frac{1}{4}}$, with $Dom(f)=\mathbb{R}$ and $Im(f)=(0,+\mathrm{\infty })$.

Solution:

1. $b=4$, $Dom(f)=\mathbb{R}$, $Im(f)=(0,+\mathrm{\infty })$
2. $b={\displaystyle \frac{1}{5}}$, $Dom(f)=\mathbb{R}$, $Im(f)=(0,+\mathrm{\infty })$
3. $b={\displaystyle \frac{1}{4}}$, $Dom(f)=\mathbb{R}$, $Im(f)=(0,+\mathrm{\infty })$

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