i am not sure how to do these problems...
*use the one-to-one property to solve the equation for x*
1. 2^(x-2)= 1/32
2. e^(x^2-3)=e^2x
I have no idea what the one-on-one property is.
Both of these log equations can be solved by taking the log of each side
1- take log2 of each side,
x-2=-5
2- take ln of each side
x^2-3=2x
the 1-1 property states that if the bases are equal the exponents are equal. therefore u must set 1/32 as 2^1/5. then 2^(x-2)= 2^(1/5) and then x-2=1/5. x=11/5
for 2), same thing. x^2 - 3= 2x.
x^2 - 2x - 3 = 0 (x-3)(x+1) = 0 x=3,-1.
how to solve 5^x-2 = 1/125
e^(X^2+6)=e^5X
No problem! I can help you with that.
To use the one-to-one property to solve equations, we need to find the inverse function of the base. In these examples, the base of the first equation is 2, and the base of the second equation is e.
Let's walk through each problem step by step:
1. 2^(x-2) = 1/32
To solve this equation using the one-to-one property, we need to take the logarithm with the same base on both sides. In this case, we take the logarithm with base 2.
Step 1: Take the logarithm of both sides using base 2:
log₂(2^(x-2)) = log₂(1/32)
Step 2: Simplify using the logarithm property:
(x - 2)log₂(2) = log₂(1/32)
Step 3: Simplify further:
x - 2 = log₂(1/32)
Step 4: Evaluate log₂(1/32):
x - 2 = -5
Step 5: Solve for x:
x = -5 + 2
x = -3
So, the solution to the equation 2^(x-2) = 1/32 is x = -3.
Now let's move on to the second problem:
2. e^(x^2-3) = e^(2x)
In this equation, the base is e. Again, we can solve it using the one-to-one property by taking the natural logarithm (ln) on both sides.
Step 1: Take the natural logarithm (ln) of both sides:
ln(e^(x^2-3)) = ln(e^(2x))
Step 2: Simplify using the logarithm property:
(x^2-3)ln(e) = (2x)ln(e)
Step 3: Simplify further (since ln(e) = 1):
x^2 - 3 = 2x
Step 4: Rearrange the equation:
x^2 - 2x - 3 = 0
Step 5: Solve for x:
This equation can be factored as (x-3)(x+1) = 0, giving the two possible solutions:
x-3 = 0, which implies x = 3
x+1 = 0, which implies x = -1
So, the solutions to the equation e^(x^2-3) = e^(2x) are x = 3 and x = -1.
I hope that helps! Let me know if you have any further questions.