Solve 2^3-x = 5^2x+1
Give an exact solution.
do you mean
2^(3-x) = 5^(2x+1) ??????
if so then
(3x-1) ln 2 = (2x+1) ln 5
3 x ln 2 - 2 x ln 5 = ln 5 + ln 2
x (ln 8 - ln 25) = ln 10
x ln (8/25) = ln 10
x = ln 10 / ln (8/25)
To solve the equation 2^(3-x) = 5^(2x+1) for an exact solution, we can rewrite the equation using the property that a^(b+c) = a^b * a^c.
Rewriting the equation, we have:
2^3 * 2^(-x) = 5^2 * 5^(2x)
Next, simplify the left side of the equation:
8 * 2^(-x) = 25 * 5^(2x)
Now, let's simplify the equation further by expressing both sides with the same base. Since 2 can be written as 5/5, we have:
(5/5)^3 * (5/5)^(-x) = 25 * 5^(2x)
Using the property (a/b)^c = a^c / b^c, we can rewrite the equation as:
(5^3 / 5^x) * (5^ (-x)) = 25 * 5^(2x)
Now, simplify the numerator:
5^3 * 5^(-x) * 5^(-x) = 25 * 5^(2x)
Using the property a^m * a^n = a^(m+n), simplify the equation further:
5^(3 - x - x) = 25 * 5^(2x)
Simplifying the exponent:
5^(3 - 2x) = 25 * 5^(2x)
Since both sides of the equation have the same base, we can equate the exponents:
3 - 2x = 2 + log5(25) + x
We can simplify further by calculating log5(25):
3 - 2x = 2 + log5(5^2) + x
3 - 2x = 2 + 2 + x
Rearranging the terms:
-2x - x = 2 + 2 - 3
-3x = 1
Divide both sides by -3:
x = -1/3
Therefore, the exact solution to the equation 2^(3-x) = 5^(2x+1) is x = -1/3.