Question

How do you solve `3^(2-x)=5^(2x+1)`?

Answer 1

Exponential rules and use of power rule of logarithm can help solve such a problem easily. The step by step working with explanation is given below.

Explanation:

`3^(2-x)=5^(2x+1)`
Splitting

`3^2*3^-x = 5^(2x)*5^1` Note: `a^(m+n)=a^m * a^n`
`9*3^-x=(5^2)^x *5` Note : `(a^(mn))=(a^m)^n`

`9/3^x = 25^x * 5` Note : `a^(-m)=1/a^m`

Solving for x by isolating the term containing x to one side of the equation.

Multiply by `3^x` on both sides we get.

`9 = 25^x * 5* 3^x`

Dividing by 5 on both sides we get.

`9/5 = 25^x * 3^x`
`9/5 = (25*3)^x`
`9/5 = (75)^x`

Taking log on both the sides

`log(9/5) = log(75)^x`
`log(9/5) = xlog(75)`

Dividing `log(75)` on both the sides.

`log(9/5)/log(75) = x`

`x=log(9/5)/log(75)` answer

Note: The answer can be represented in many ways and that would be decided by the question. If a numerical value is needed then please use a calculator to find it.

`x=log_75(9/5)` can be one answer as well. Make a choice depending on the question.