The alternative method to solve the exponential equation is to form matching bases both sides such that:
Notice that when you have matching bases, you only need to equate the exponents.
We have 2^x = 16
To solve, first e will rewrite the numbers so the bases are equal.
We will factor 16.
==> 16 = 4*4 = 2*2*2*2 = 2^4
Now we will substitute into the equation.
==> 2^x = 2^4
Now we notice that the bases are equal, then the exponents must be equal too.
Then, we conclude that x = 4.
To check, we will substitute with x=4.
==> 2^x = 16
==> 2^4 = 16
==> 16 =16
We know that for solving exponential equations without the help of logarithms, we must have comparable bases both sides of the equal sign.
The assumption you've made is right, because 16 is a power of 2.
So, you have to write 16 as a power of 2, in order to get comparable bases.
2^x = 2^4
The exponential are equals if only the exponents are equals.
x = 4
You also could have:
3^(2x - 1) = 3^x
2x - 1 = x
x - 1 = 0
x = 1