BigO of Algorithm Using Multiple Variable Regression - r

Question

For more verbose algorithms, determining the time complexity (i.e. BigO) is a pain. My solution has been to time the execution of the algorithm with parameters n and k, and come up with a function (time function) that varies with n and k.
My data looks something like the below:
n k executionTime
500 1 0.02
500 2 0.03
500 3 0.05
500 ... ...
500 10 0.18
1000 1 0.08
... ... ...
10000 1 9.8
... ... ...
10000 10 74.57
I've been using the lm() function in the stats R package. I don't know how to interpret the output of the multiple regression, to determine a final Big-O. This is my main question: how do you translate the output of a multiple variable regression, to a final ruling on the best Big-O time complexity rating?
Here's the output of the lm():
Residuals:
Min 1Q Median 3Q Max
-14.943 -5.325 -1.916 3.681 31.475
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -2.130e+01 1.591e+00 -13.39 <2e-16 ***
n 4.080e-03 1.953e-04 20.89 <2e-16 ***
k 2.361e+00 1.960e-01 12.05 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 7.962 on 197 degrees of freedom
Multiple R-squared: 0.747, Adjusted R-squared: 0.7444
F-statistic: 290.8 on 2 and 197 DF, p-value: < 2.2e-16
Here's the output of log(y) ~ log(n) + log(k):
Residuals:
Min 1Q Median 3Q Max
-0.4445 -0.1136 -0.0253 0.1370 0.5007
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -16.80405 0.13749 -122.22 <2e-16 ***
log(n) 2.02321 0.01609 125.72 <2e-16 ***
log(k) 1.01216 0.01833 55.22 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1803 on 197 degrees of freedom
Multiple R-squared: 0.9897, Adjusted R-squared: 0.9896
F-statistic: 9428 on 2 and 197 DF, p-value: < 2.2e-16
Here's the output of the principle components, showing both n and k are contributing to the spread of the multivariate model:
PC1(This is n) PC2 (this is k) PC3 (noise?)
Standard deviation 1.3654 1.0000 0.36840
Proportion of Variance 0.6214 0.3333 0.04524
Cumulative Proportion 0.6214 0.9548 1.00000