If x+ ₁₅ (5+3^x )=x ₁₅ 5+ ₁₅ 24 , then x= . .

Step-by-step solution

Solution: The equation x+ ₁₅(5+3^x)=x ₁₅5+ ₁₅24 is solved using logarithmic identities. Rewriting x as ₁₅15^x and x ₁₅5 as ₁₅5^x transforms the equation to: ₁₅15^x+ ₁₅(5+3^x)= ₁₅5^x+ ₁₅24 Combining logarithms on each side gives: ₁₅ [15^x(5+3^x) ]= ₁₅(5^x 24) Equating the arguments produces: 15^x(5+3^x)=5^x 24 Dividing through by 5^x and simplifying yields 3^x(5+3^x)=24 . Substituting y=3^x leads to the quadratic y^2+5y-24=0 , which factors as (y+8)(y-3)=0 . The solution y=-8 is extraneous since 3^x>0 . The valid so