7^1_x =4^3x+1?

To solve this equation, we need to first simplify both sides of the equation.

7^1_x = 7^x

4^3x+1 = 4^(3x+1)

Now, we can set the two expressions equal to each other:

7^x = 4^(3x+1)

To solve for x, we can take the natural logarithm of both sides:

ln(7^x) = ln(4^(3x+1))

Using the property of logarithms that ln(a^b) = b*ln(a), we can simplify the equation further:

x*ln(7) = (3x+1)*ln(4)

Now, we can expand both sides of the equation:

x*ln(7) = 3x*ln(4) + ln(4)

Next, we can isolate x on one side of the equation:

x*ln(7) - 3x*ln(4) = ln(4)

Factor out x:

x(ln(7) - 3ln(4)) = ln(4)

Divide by (ln(7) - 3ln(4)) to solve for x:

x = ln(4) / (ln(7) - 3ln(4))

Therefore, x = ln(4) / (ln(7) - 3ln(4)) is the solution to the equation 7^x = 4^(3x+1).