Solve equation using exact values

Question

e^3x=5^x-1

Answer 1

We can rewrite 5^x-1 as     e^{ln(5^(x + 1))}

 

e^3x = e^{ln(5^(x - 1))}

 

 

Equate exponents.

 

3x = ln(5^{x - 1})

 

 

Bring the exponent as coefficients of logs.

 

3x = (x - 1)ln5

 

 

Distribute ln5.

 

3x = xln5 - ln5

 

 

Isolate all x terms to one side.

 

3x - xln5 = -ln5

 

x(3 - ln5) = -ln5

 

x = -ln5 / (3 - ln5)

 

This value of x is exact value.