Solve the exponential equation.
4^2x – 2 = 26
I know I use the one to one property to equate. But I am confused. Thanks!
do you mean
4^(2x) - 2 = 26 or
4^(2x-2) = 26 or
(4^2)x - 2 = 26
The way you typed it, it would be the last one, but I am sure that is not what you meant.
4^(2x-2)=26
then take log of both sides
log [4^(2x-2) }= log 26
(2x-2)log4 = log 26
2x-2 = log26/lo4
2x = (log26/log4 + 2)
x = (log26/log4 + 2)/2 = appr. 2.1751
check
4^(2(2.1751 - 2)
= 4^(4.35022 - 2)
= 4^2.35022
= 26
= RS
To solve the exponential equation 4^(2x – 2) = 26, we can follow these steps:
Step 1: Apply the one-to-one property of exponential functions. This property states that if two exponential expressions with the same base are equal, then their exponents must be equal as well.
Step 2: Rewrite 26 as a power of 4. We want to rewrite 26 as 4 raised to some exponent.
Since 4^2 = 16 and 4^3 = 64, we can see that 26 falls between the two powers of 4. Therefore, we can approximate 26 as 4^(2 + 1/2), which is equal to 4^2 * 4^(1/2).
We can rewrite 26 as 4^(2 + 1/2) = 4^2 * 4^(1/2). Therefore, our equation becomes:
4^(2x – 2) = 4^2 * 4^(1/2)
Step 3: Apply the rules of exponents. When we have the same base raised to different exponents, we can use the rule a^(m+n) = a^m * a^n. Applying this rule, we have:
4^(2x – 2) = 4^(2 + 1/2)
Step 4: Set the exponents equal to each other. Since the bases are the same, we can equate the exponents:
2x – 2 = 2 + 1/2
Step 5: Solve for x. Now, we can solve for x by isolating it on one side of the equation:
2x = 2 + 1/2 + 2
2x = 4 + 1/2
2x = 8/2 + 1/2
2x = 9/2
x = (9/2) / 2
x = 9/4
Therefore, the solution to the exponential equation 4^(2x – 2) = 26 is x = 9/4.