Solve the exponential equation (3/5)^x=7^(1-x) without using a calculator?

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The equation `(3/5)^x=7^(1-x)` has to be solved.

`(3/5)^x=7^(1-x)`

Use the formula `a^(b - c) = a^b/a^c`

=> `(3/5)^x = 7/7^x`

Use the formula `a^x*b^x = (a*b)^x`

=> `((3*7)/5)^x = 7`

=> `(21/5)^x = 7`

To solve further take the logarithm of both the sides.

`log((21/5)^x) = log 7`

Use the formula `log a^x = x*log a`

=> `x*log(21/5) = log 7`

=> `x = (log 7)/(log(21/5))`

The solution of the equation `(3/5)^x=7^(1-x)` is `x = (log 7)/(log(21/5))`

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aruv | High School Teacher | (Level 2) Valedictorian

Posted on

Solve the exponential equation (3/5)^x=7^(1-x).

`(3/5)^x=7^(1-x)`

Taking log both sides

`log(3/5)^x=log7^(1-x)`

`xlog(3/5)=(1-x)log7`

`xlog(3/5)=log7-xlog7`

`xlog(3/5)+xlog7=log7`

`x(log3-log5+log7)=log7`

`x=log7/(log3+log7-log5)`

In above we have used the following rules in logarithm.

`1. log(x/y)=logx-logy`

`2. logx^m=mlogx`

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