Solve for x 5^(3x+1)=3^(x-5)

Problem

5^(3*x+1)=3^(x−5)

Solution

1. Take the natural logarithm of both sides of the equation to move the variables out of the exponents.

ln(5^(3*x+1))=ln(3^(x−5))

2. Apply the power rule for logarithms, ln(a^b)=b*ln(a) to both sides.

(3*x+1)*ln(5)=(x−5)*ln(3)

3. Distribute the logarithmic constants into the binomial terms.

3*x*ln(5)+ln(5)=x*ln(3)−5*ln(3)

4. Isolate the terms containing x on one side of the equation and the constants on the other.

3*x*ln(5)−x*ln(3)=−5*ln(3)−ln(5)

5. Factor out the common factor x from the left side.

x*(3*ln(5)−ln(3))=−5*ln(3)−ln(5)

6. Solve for x by dividing both sides by the coefficient (3*ln(5)−ln(3))

x=(−5*ln(3)−ln(5))/(3*ln(5)−ln(3))

7. Simplify the expression using logarithm properties if desired, such as n*ln(a)=ln(a^n) and ln(a)+ln(b)=ln(a*b)

x=(−(ln(3^5)+ln(5)))/(ln(5^3)−ln(3))

x=(−ln(243⋅5))/(ln(125)−ln(3))

x=(−ln(1215))/ln(125/3)

Final Answer

x=(−5*ln(3)−ln(5))/(3*ln(5)−ln(3))

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