Approximating e^x

 Approximating exponentials manually. 

What is the value of \( e^{60} \) ? or \( e^{1000} \)? Most calculators show Math error. Here is a way to calculate manually.  

Let \( x = e^{60} \). Taking \( log_{10} \) on both sides we get, \[ log_{10}x = 60/2.303 ≈ 26.05 \] So, let \( x = x' \times 10^{26} \). Now substituting this and taking \( ln \) we get \[ ln x' + 2.303 \times 26 = 60 \implies x' = e^{0.122} \]. When \( x< 1, e^x ≈ 1 + x \). So, \( x' = 1+0.122 = 1.122 \) , therefore, \( e^{60} ≈ 1.12 \times 10^{26} \) 

 Now, let's find \( e^{1000} \). Let \( x = e^{1000} \). \[ log_{10}x = 1000/2.303 ≈ 434.21 \] So let \( x = x' \times 10^{434} \). \[ ln x' + 434 \times 2.303 = 1000 \implies x = e^{0.498} ≈ 1 + 0.498 \] So, \( e^{1000} ≈ 1.5 \times 10^{434} \).

Note that \( e^x \) when \( x \in (1,2) = e^{1+(x-1)} ≈ 2.7 \times e^{x-1} ≈ 2.7 x \).