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.