Calculating very large powers of $e$

Question

I need to calculate $e^{1763.192674118048}$, but when I try to calculate it directly using Matlab it returns "Inf", i.e. it can't calculate it. How do I calculate this? For what it's worth, just getting the right power of 10 would be accurate enough for my purposes.

Answer 1

$$ \left\lfloor \frac{1763.192674118048}{\log 10} \right\rfloor = 765 $$ This is the logarithm base $e,$ so $\log 10 \approx 2.30258509$

Since $$ \frac{1763.192674118048}{\log 10} \approx 765.7448488 $$ we find that your number is $$ e^{1763.192674118048} \approx 5.5571 \cdot 10^{765} $$ because $$ 10^{0.7448488} \approx 5.5571 $$

Answer 2

To get the approximate power of ten, i.e. the $\alpha$ in $e^x \approx 10^\alpha$, by taking natural logarithms on both sides, $x \approx \alpha \ln 10$, so $\alpha = \frac x {\ln 10}$. That gives approximately $765$.

Answer 3

If, for your purposes, it would be sufficient to convert a power of e into a power of 10, then you can just change the base of the exponent:

$$e^x = (10^{\log_{10}(e)})^x = 10^{x \log_{10}(e)}$$

For example, this is done in the following Matlab code:

x = 3.2
y = x*log10(exp(1))
exp(x), 10^y

Answer 4

Use vpa (variable-precision arithmetic). You can do it with strings

>> vpa('exp(1763.192674118048)')
ans =
5.5571088254929495883970009541213*10^765

or defining a symbolic variable

>> x = sym(1763.192674118048);
>> vpa(exp(x))
ans =
5.5571088254928906583892856815215*10^765

Answer 5

Since your looking for the power of 10, what you're asking is this:

Find $y$ such that $$e^{1763.192674118048}=10^y$$

Take the natural log of both sides:

$$\ln e^{1763.192674118048}= \ln 10^y$$

Bring the exponents to the front:

$$1763.192674118048\ln e= y\ln 10$$

Rewrite in terms of $y$ and evaluate ($\ln e = 1$):

$$y=\frac {1763.192674118048 \ln e}{\ln 10}= \frac{1763.192674118048}{\ln 10}\approx 765.7448489017067943$$

While $765.7448489017067943$ is number you're looking for because $$e^{1763.192674118048}\approx 10^{765.7448489017067943}$$
you could write this in spiffy standard notation. Consider:

$$10^{765.7448489017067943}=(10^{765})(10^{0.7448489017067943})$$

$$=5.5571088254929496998\times 10^{765}$$

And, yes, that is a lot of decimal points, but we were fed a lot to begin with, so it's a fair trade.