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

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)  
------------------------------------------------------------------------------------------
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)
--------------------------------------------------------------------------
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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