We can start by simplifying the left-hand side of the equation:

1^x + 2^x = 1 + 2^x

Now we can substitute this

contexto expression back into the original equation to get:

1 + 2^x = 4^x

We can rewrite 4 as 2^2, and then apply the exponent rules for products to get:

1 + 2^x = (2^2)^x = 2^(2x)

Now we have an equation with only one base, so we can use logarithms to solve for x:

log₂(1 + 2^x) = log₂(2^(2x))

log₂(1 + 2^x) = 2x

Now we can use the change-of-base formula to rewrite the left-hand side in terms of natural logarithms:

ln(1 + 2^x) = 2x / ln(2)