exponential

Question

2^x=16^x-1

Answer 1

Kenneth is right: 2^x = 16^x - 1 has a closed form solution, but it is messy, requiring the procedure for solving a general 4th degree polynomial:

 

https://www.wyzant.com/resources/files/372247/solving_general_cubic_and_quartic_equations

 

Instead you must have meant 2^x=16^x-1, which has a simple solution by using 16=2⁴

 

2^x = 2^4(x-1)

 

4(x-1) = x

 

4x - 4 = x

 

3x = 4

 

x = 4/3

 

Answer 2

Replace 16 by 2⁴:

2^x = (2⁴)^x-1   NOTE: It's necessary to assume that the "x-1" is the entire exponent on the right-side of original equation; technically, your should have enclosed that entire exponent in parentheses.

 

Since these two exponentials have the same base (2), you now can EQUATE THE EXPONENTS; therefore

x = 4(x-1)

x = 4x - 4

4 = 3x

x = 4/3

 

Check this answer on calculator, being sure to use sufficient parentheses to obtain the correct order of operations.