`3^(2x) = 75` Solve the equation accurate to three decimal places

Problem:` 3^(2x)=75` is an exponential equation.

To simplify, we need to apply logarithm property: `log(x^y) = y*log(x)`

to bring down the exponent that is in terms of x.

Taking "log" on both sides:

`log(3^(2x))=log(75 )`

`(2x)log(3)=log(75)`

Divide both sides by `log(3)` to isolate "`2x` ":

`(2x * log (3)) /(log(3))= (log(75))/(log(3))`

`2x=(log(75))/(log(3))`

Multiply both sides by 1/2 to isolate x:

`(1/2)*2x=(log(75))/(log(3))*(1/2)`

Note: You will get the same result when you divide both sides by 2.

The equation becomes:

`x=(log(75))/(2log(3))`

`x~~1.965` Rounded off to three decimal places

To check, plug-in `x=1.965` in` 3^(2x)=75` :

`3^(2*1.965)=?75`

`3^(3.93)=?75`

`75.0043637 ~~75` TRUE

Conclusion: `x~~1.965` is the final answer.

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