To solve for x without using a calculator, we can start by applying the properties of logarithms. First, we need to simplify the equations on both sides of the equation.
1. Start with the equation: log5(3^x) = log25(9^(1-2x))
2. Use the property of logarithms loga(b^c) = c * loga(b) to simplify the equation:
x * log5(3) = (1 - 2x) * log25(9)
3. Use the property loga(b^c) = c * loga(b) again on the right side:
x * log5(3) = (1 - 2x) * (log3(9)/log3(5^2))
4. Simplify the logarithms on the right side:
x * log5(3) = (1 - 2x) * (2*log3(3)/2*log3(5))
x * log5(3) = (1 - 2x) * (log3(3)/log3(5))
5. Simplify further:
x * log5(3) = (1 - 2x)
6. Distribute on the right side:
x * log5(3) = 1 - 2x
7. Move all the terms to one side of the equation:
x * log5(3) + 2x = 1
8. Factor out the x:
x * (log5(3) + 2) = 1
9. Divide both sides by (log5(3) + 2):
x = 1 / (log5(3) + 2)
Now you have solved for x without using a calculator.