`10^(3x-8)=2^(5-x)` Solve the equation.

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marizi eNotes educator| Certified Educator

To solve the equation: `10^(3x-8)=2^(5-x)` , we may take "ln" on both sides.

`ln(10^(3x-8))=ln(2^(5-x))`

Apply natural logarithm property: `ln(x^n) = n*ln(x)` .

`(3x-8)ln(10)=(5-x)ln(2)`

Let `10=2*5` .

`(3x-8)ln(2*5)=(5-x)ln(2)`

Apply natural logarithm property: `ln(x*y) = ln(x)+ln(y)` .

`(3x-8)(ln(2) +ln(5))=(5-x)ln(2)`

Distribute to expand each side.

`3xln(2) +3xln(5)-8ln(2) -8ln(5)=5ln(2)-xln(2)`

Isolate all terms with x's on one side.

`3xln(2) +3xln(5)-8ln(2) -8ln(5) =5ln(2)-xln(2)`

                                  `+8ln(2) +8ln(5) `     `+8ln(2) `        ` +8ln(5)`  

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`3xln(2)+3xln(5)+0 +0 =13ln(2)-xln(2) +8ln(5)`

 

`3xln(2)+3xln(5) =13ln(2)-xln(2) +8ln(5)`

`+xln(2) `                      ` +xln(2)`

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`4xln(2) +3xln(5) =13ln(2)-0+8ln(5)`

`4xln(2) +3xln(5) =13ln(2)+8ln(5)`

Factor out common factor `x` on the left side.

 

`x(4ln(2) +3ln(5)) =13ln(2)+8ln(5)`

Divide both sides by `(4ln(2) +3ln(5))` .

`(x(4ln(2) +3ln(5)))/(4ln(2) +3ln(5)) =(13ln(2)+8ln(5))/(4ln(2) +3ln(5))`

`x=(13ln(2)+8ln(5))/(4ln(2) +3ln(5))`

Apply natural logarithm property: `n*ln(x)=ln(x^n)`

`x=(ln(2^(13))+ln(5^8))/(ln(2^4) +ln(5^3))`

`x=(ln(8192)+ln(390625))/(ln(16) +ln(125))`

Apply natural logarithm property: `ln(x)+ln(y)=ln(x*y)` .

`x=(ln(8192*390625))/(ln(16*125))`

`x=(ln(3200000000))/(ln(2000))`

or

`x~~2.879`

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