Solve the following exponential equation
5^2-x=7^5x+7
5 ^ 2 = 25
7 ^ 5 = 16,807
5 ^ 2 - x = 7 ^ 5 x + 7
25 - x = 16,807 x + 7 Add x to both sides
25 - x + x = 16,807 x + 7 + x
25 = 16,808 x + 7 Subtract 7 to both sides
25 - 7 = 16,808 x + 7 - 7
18 = 16,808 x Divide both sides by 16,808
18 / 16,808 = 16,808 x / 16,808
18 / 16,808 = x
2 * 9 / ( 2 * 8,404 ) = x
9 / 8,404 = x
x = 9 / 8,404
If you mean,
5^(2-x) = 7^(5x + 7)
Then, we take the ln of both sides
(2-x)*ln(5) = (5x + 7)*ln(7)
1.609*(2-x) = 1.946*(5x+7)
Solving for x,
x = -0.9174
Hope this helps :3
To solve the given exponential equation 5^(2-x) = 7^(5x+7), we can take the natural logarithm (ln) of both sides of the equation. This is because the natural logarithm is the inverse function of the exponential function. Applying the natural logarithm will allow us to simplify the equation and solve for the variable.
Taking the natural logarithm of both sides gives:
ln(5^(2-x)) = ln(7^(5x+7))
According to logarithm rules, we can simplify the equation by bringing down the exponents as coefficients:
(2-x)ln(5) = (5x+7)ln(7)
Next, distribute the coefficients:
2ln(5) - xln(5) = 5xln(7) + 7ln(7)
Now, we can rearrange the equation by combining the terms with "x" on one side and the constant terms on the other side. This will help us isolate "x":
2ln(5) - 7ln(7) = 5xln(7) + xln(5)
Rearranging further:
2ln(5) - 7ln(7) = xln(5) + 5xln(7)
Factoring out common factors on both sides:
(2-ln(5)) - 7ln(7) = x(ln(5) + 5ln(7))
Finally, we divide both sides of the equation by (ln(5) + 5ln(7)) to solve for "x":
x = (2-ln(5) - 7ln(7)) / (ln(5) + 5ln(7))
Calculating the numeric value of "x" is straightforward once you substitute the values of ln(5) and ln(7) with their decimal approximations.