You can't really combine the assets' log returns. You should calculate percentage returns for the three assets. Then at each time step, the portfolio's total return is:
$r(i) = 0.5 \times \text{asset1_return}(i) + 0.25 \times \text{asset2_return}(i) + 0.25 \times \text{asset3_return}(i)$
Once you've calculated the time series of the portfolio's returns, you create another time series by adding 1 to each return, $r(i)$ -- let's call this series "$s$". It is the daily return of a portfolio with 50% in asset 1, 25% in asset 2 and 25% in asset 3.
Then the portfolio index is created by multiplying each successive element of "$s$" with all the previous elements. This can be written recursively as:
$\text{Index}(i) = \text{Index}(i-1) \times s(i)$
This index then represents what a daily rebalanced portfolio would perform as.
Importantly, the index weights are constant, so if you had \$100 invested the first day, the portfolio would consist of \$50 in the first asset and \$25 each in assets 2 and 3. If on the second day, the portfolio value was \$103, then the procedure above assumes that the portfolio is rebalanced to consist of \$51.50 in the first asset and \$25.75 in the second and third assets.
Note that I've just asked some questions about this procedure elsewhere on this site -- this is the classic way to backtest a strategy but in reality few portfolios rebalance daily like this.