A pre-loaded example dataset in R

Main page: https://www.rdocumentation.org/packages/datasets/versions/3.6.2/topics/trees

trees
##    Girth Height Volume
## 1    8.3     70   10.3
## 2    8.6     65   10.3
## 3    8.8     63   10.2
## 4   10.5     72   16.4
## 5   10.7     81   18.8
## 6   10.8     83   19.7
## 7   11.0     66   15.6
## 8   11.0     75   18.2
## 9   11.1     80   22.6
## 10  11.2     75   19.9
## 11  11.3     79   24.2
## 12  11.4     76   21.0
## 13  11.4     76   21.4
## 14  11.7     69   21.3
## 15  12.0     75   19.1
## 16  12.9     74   22.2
## 17  12.9     85   33.8
## 18  13.3     86   27.4
## 19  13.7     71   25.7
## 20  13.8     64   24.9
## 21  14.0     78   34.5
## 22  14.2     80   31.7
## 23  14.5     74   36.3
## 24  16.0     72   38.3
## 25  16.3     77   42.6
## 26  17.3     81   55.4
## 27  17.5     82   55.7
## 28  17.9     80   58.3
## 29  18.0     80   51.5
## 30  18.0     80   51.0
## 31  20.6     87   77.0
require(stats); require(graphics)
pairs(trees, panel = panel.smooth, main = "trees data")

plot(Volume ~ Girth, data = trees, log = "xy")

coplot(log(Volume) ~ log(Girth) | Height, data = trees,
       panel = panel.smooth)

summary(fm1 <- lm(log(Volume) ~ log(Girth), data = trees))
## 
## Call:
## lm(formula = log(Volume) ~ log(Girth), data = trees)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.205999 -0.068702  0.001011  0.072585  0.247963 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -2.35332    0.23066  -10.20 4.18e-11 ***
## log(Girth)   2.19997    0.08983   24.49  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.115 on 29 degrees of freedom
## Multiple R-squared:  0.9539, Adjusted R-squared:  0.9523 
## F-statistic: 599.7 on 1 and 29 DF,  p-value: < 2.2e-16
summary(fm2 <- update(fm1, ~ . + log(Height), data = trees))
## 
## Call:
## lm(formula = log(Volume) ~ log(Girth) + log(Height), data = trees)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.168561 -0.048488  0.002431  0.063637  0.129223 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -6.63162    0.79979  -8.292 5.06e-09 ***
## log(Girth)   1.98265    0.07501  26.432  < 2e-16 ***
## log(Height)  1.11712    0.20444   5.464 7.81e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.08139 on 28 degrees of freedom
## Multiple R-squared:  0.9777, Adjusted R-squared:  0.9761 
## F-statistic: 613.2 on 2 and 28 DF,  p-value: < 2.2e-16
step(fm2)
## Start:  AIC=-152.69
## log(Volume) ~ log(Girth) + log(Height)
## 
##               Df Sum of Sq    RSS      AIC
## <none>                     0.1855 -152.685
## - log(Height)  1    0.1978 0.3832 -132.185
## - log(Girth)   1    4.6275 4.8130  -53.743
## 
## Call:
## lm(formula = log(Volume) ~ log(Girth) + log(Height), data = trees)
## 
## Coefficients:
## (Intercept)   log(Girth)  log(Height)  
##      -6.632        1.983        1.117
## i.e., Volume ~= c * Height * Girth^2  seems reasonable