You can easily double the investment and also double your return right? Not really, because you are doubling the risk as well… read on to see why.
Returns
Singleperiod returns
Simple return
Also called linear or raw return is defined as: \[ \color{blue} { r_t = \frac{x_t  x_{t1}}{x_{t1}} = \frac{x_t}{x_{t1}}  1 } \] where:

r_{t}  simple return for period t

x_{t}  current value (price)

x_{t1}  previous value (price)
Log return
Also called compound return is defined as: \[ \color{blue} { R_t = \ln({\frac{x_t}{x_{t1}}}) = \ln({x_t})  \ln{(x_{t1})} } \] where:

R_{t}  log return for period t

ln  natural logarithm (base e  Euler's number)
Multiperiod returns
Simple return
Simple total return is defined as sum of individual returns for each period: \[ \color{blue} { str_t^n = {\sum_{t=1}^n{r_t}} } \] while simple average return is just the arithmetic mean: \[ \color{blue} { sar_t^n = \frac{\sum_{t=1}^n{r_t}}{n} } \] Example 1: let's take a stock A that is initially worth $100 and rises to $120 in t_{1} and goes back to $100 in t_{2}
a0, a1, a2 = 100, 120, 100
r1_a = a1/a0  1
print(r1_a)
r2_a = a2/a1  1
print(r2_a)
str_a = r1_a + r2_a
print(str_a)
sar_a = tr / 2
print(sar_a)
0.19999999999999996 0.16666666666666663 0.033333333333333326 0.016666666666666663
Ooops! Something does not look right, total return is 3.33% even if we did not make any money (price falls back to $100).
Example 2: Let's take another stock B that was trading at $100 then $140 and $145.
b0, b1, b2 = 100, 140, 145
r1_b = b1/b0  1
print(r1_b)
r2_b = b2/b1  1
print(r2_b)
0.3999999999999999 0.03571428571428581
Now, let's suppose that we have a portfolio with stock A (weight: 0.6) and stock B (weight: 0.4) and we want to calculate portfolio return for each period.
r1 = 0.6 * r1_a + 0.4 * r1_b
print(r1)
r2 = 0.6 * r2_a + 0.4 * r2_b
print(r2)
0.2693929340763724 0.09510721979065809
Good! Return per each period seems correct given the weight of each stock.
Conclusions: simple returns

ARE NOT additive over time

ARE additive over assets
Log return
Compound total return is the sum of all log returns: \[ \color{blue} { ctr_t^n = {\sum_{t=1}^n{R_t}} } \] and for compound average return we need to use geometric mean because arithmetic mean does not account for compounding: \[ \color{blue} { car_t^n = ({\prod_{t=1}^n{(1 + R_t)}})^{\frac{1}{n}}  1 } \]
Example: let's use the same example above and calculate the returns:
from math import log
r1_a = log(a1)  log(a0)
print(r1_a)
r2_a = log(a2)  log(a1)
print(r2_a)
ctr = r1_a + r2_a
print(ctr)
car = ((1+r1_a) * (1+r2_a))**(1/2)  1
print(car)
0.182321556793954 0.182321556793954 0.0 0.016761041288421596
Good! compound total return is 0% which is the correct result and compound average return seems to be correct and equal to simple average return 1.66%.
Example 2: same stock B as above
r1_b = log(b1)  log(b0)
print(r1_b)
r2_b = log(b2)  log(b1)
print(r2_b)
0.33647223662121206 0.03509131981127034
And portfolio return per each period:
r1 = 0.6 * r1_a + 0.4 * r1_b
print(r1)
r2 = 0.6 * r2_a + 0.4 * r2_b
print(r2)
0.24398182872485724 0.09535640615186428
Ooops! quite a difference between simple (0.27) and compound (0.24) portfolio return for t_{1} because when we add log returns we compound but in this case there is nothing to compound on, stock A and stock B are two different things which may/may not be correlated.
Conclusions: log returns

ARE additive over time

ARE NOT additive over assets
Simple vs. log return
OK, both returns have pros/cons but which one is better and the answer is both, it depends on what kind of data you have and what you want to calculate.
A few more properties and intuition:

log return as function of simple return
\[ \textcolor{black} { R_t = \ln({\frac{x_t}{x_{t1}}}) \\ R_t = \ln({\frac{x_t}{x_{t1}}}  1 + 1) } \] \[ \color{blue} { R_t = \ln(r_t + 1) } \]

easy to calculate one given the other
\[ \textcolor{black} { e^{R_t} = e^{\ln(r_t + 1)} \\ e^{R_t} = r_t + 1 } \] \[ \color{blue} { r_t = e^{R_t}  1 } \]

aggregation (sum) is very efficient over log returns, a sub operation executed in O(1) \[ \textcolor{black} { R = \ln(\frac{x_f}{x_{f1}}) + ... + \ln(\frac{x_1}{x_0}) \\ R = \ln(\frac{x_f}{x_{f1}}\frac{x_{f1}}{x_{f2}}...\frac{x_2}{x_1}\frac{x_1}{x_0}) \\ R = \ln({\frac{x_f}{x_0}}) } \]
\[ \color{blue} { R = ln(x_f)  ln(x_0) } \] where:

R  total return

x_{f}  final value (price)

x_{0}  initial value (price)


log return is faster to calculate because subtraction operation are numerically safe/faster than division

log returns follow normal distribution since underlying prices are (hopefully) lognormal distributed (see Central Limit Theorem)

simple returns are easier to reason about

if you work with raw data (prices) use simple returns and arithmetic mean

when you work with percentage/change in values use log returns

if data series are volatile use log returns and geometric mean

aggregate data over time use log returns

aggregate data over different assets use simple returns
Riskadjusted ratios
Now, with returns out of the way, let's get back to our problem.
Why doubling the investment is not a good idea, because you also doubling the risk, that's why we need to use riskadjusted return (ratio) as the ones presented below.
Sharpe ratio
Sharpe Ratio is defined as excess return (portfolio return rate  riskfree rate) divided by volatility.
\[ \textcolor{blue} { Sharpe\ ratio = \frac{r_p  r_f}{\sigma} } \] where:

r_{p}  average portfolio return

r_{f}  riskfree rate

σ  standard deviation (both upside and downside volatility)
Sortino ratio
This is very useful to investors and longonly traders/funds to calculate the return for a given bad risk (downside only volatility), because they love upside volatility but hate downside risk.
\[ \textcolor{blue} { Sortino\ ratio = \frac{R_p  r_t}{\sigma_d} } \] where:

R_{p}  compound (realized) portfolio return

r_{t}  target return

σ_{d}  downside volatility
Information ratio
\[ \textcolor{blue} { Information\ ratio = \frac{R_p  r_b}{\sigma_a} } \] where:

R_{p}  portfolio return

r_{b}  benchmark return

σ_{a}  volatility of active returns
Other metrics
These are not riskadjusted ratios but are good to keep an eye on them.
Gain to Pain ratio
It calculates the bang for the buck ratio, the amount of loss (pain) that is needed to play the game and make some profit (gain).
\[ \textcolor{blue} { GtP\ ratio = \frac{\sum_{t=1}^n{r_t}}{abs(\sum_{t=1}^n{r_{t,n}})} } \] where:

r_{t}  return for period t

r_{t,n}  period with negative return

abs  take the absolute value
Profit factor
Similar to GtP ratio but it uses profitable periods instead of all periods.
\[ \textcolor{blue} { PF\ ratio = \frac{\sum_{t=1}^n{r_{t,p}}}{abs(\sum_{t=1}^n{r_{t,n}})} } \] where:

r_{t,p}  period with positive return
Win rate
Number of trades in profit vs. total number of trades.
\[ \textcolor{blue} { Win\ rate = \frac{\#\ of\ t_w}{\#\ of\ t_t} } \] where:

t_{w}  win trades

t_{t}  total trades
Avg win vs. loss ratio
It compares the average size of win vs. loss trades.
\[ \textcolor{blue} { Win\ /Loss = \frac{avg(\sum_{t=1}^n{r_{t,p}})}{avg(\sum_{t=1}^n{r_{t,n}})} } \] where:

avg  take the average
References

https://quantivity.wordpress.com/2011/02/21/whylogreturns/

https://assylias.wordpress.com/2011/10/27/linearvslogarithmicreturns/

https://investmentcache.com/magicoflogreturnsconceptpart1/

https://www.investopedia.com/articles/stocks/11/5waystomeasuremoneymanagers.asp

https://www.thebalance.com/calculatecompoundannualgrowthrate357621

https://www.goodreads.com/book/show/54308357unknownmarketwizards

https://www.investopedia.com/ask/answers/06/geometricmean.asp

https://moontowermeta.com/pathhowcompoundingaltersreturndistributions/