Why does the inverse of ln lead to this expression?

Question

I have the following information available

The graph for a function that describes the relation between year (t) and ln(y) is shown here

Based on the given data, a function has been derived using linear regression with the result that $$ \text{ln }y \simeq 0.507t-983.054 $$

Now I have been told that based on this information, I should be able to describe $y = y(t)$ with the expression $$y = y(t) \simeq 1.42 \cdot 10^{15} \cdot e^{0.507(t-2008)}$$ Why does the expression for $y$ look like this? I have tried to take the expontential function of both sides of $\text{ln }y \simeq 0.507t-983.054$ but I still cannot see why the expression for $y$ looks like that.

Answer 1

Note that$$1.42\cdot10^{15}\simeq e^{34.8894}$$and that therefore$$1.42\cdot10^{15}\cdot e^{0.507\times(t-2008)}\simeq e^{34.8894-0.507\times2008+0.507t}.$$But $34.8894-0.507\times2008\simeq-983.054$. So$$1.42\cdot10^{15}\cdot e^{0.507\times(t-2008)}\simeq e^{0.507t-983.054}.$$