We just started logarithms and exponential functions, and I'm kind of getting the gist but not really. Here is what I tried on one of the homework problems:
2^X=4^X+1
1)2^X=(2^2)^X+1
2)x=x+1
3)0=1 or no solutions
I wasn't sure what to do between steps 1 and 2. Thanks!!
If the question is the way you typed it, it is a nasty question.
I am pretty sure you meant
2^x = 4^(x+1)
did you notice that both bases are powers of 2 ?
2^x = (2^2)^(x+1)
2^x = 2^(2x+2)
then 2x + 2 = x
x = -2
I see where the extra step was! Thanks so much!
To solve the equation 2^X = 4^X+1, you need to use logarithms to isolate the variable X.
1) First, rewrite 4 as 2^2, so the equation becomes 2^X = (2^2)^(X+1).
This step is correct, as it allows you to rewrite the exponential equation using the same base (2).
2) Now, you need to apply the exponent properties to simplify the equation.
The property you can use here is (a^m)^n = a^(m*n). Applying it, you get 2^X = 2^(2*(X+1)).
This step combines the exponents within the parentheses.
3) Since the bases (2) are the same, you can equate the exponents:
X = 2*(X+1).
This step is where the error occurred in your solution. The exponents should be set equal to each other, not the bases.
To continue solving:
4) Distribute the 2 on the right side:
X = 2X + 2.
5) Simplify the equation:
X - 2X = 2,
-X = 2.
6) Multiply both sides by -1 to isolate X:
X = -2.
So, the solution to the equation 2^X = 4^X+1 is X = -2.