solve for x without using a calculator:

log5(3^x) = log25(9^(1-2x))

I don't know how to get rid of the logs for this one. Help?

Thank you!

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.