Problem 2018

Solve the logarithmic equation.

log₅^x + log₅^(x+2) = log₅³

Solution:-

Rewrite the left side of the equation as one logarithm, using the properties of logarithms.

log₅^(x(x+2)) = log₅³

Next, exponentiate both sides of the equation.

x(x+2) = 3

Now, solve for x. Use the Distributive Property.

x² + 2x = 3

Set the equation equal to zero.

x² + 2x -3 = 0

Solve the resulting quadratic equation. Use the Quadratic Formula.

x = $\frac{-2\pm \sqrt{16}}{2}$

Remember that the domain of log_mx is all x > 0, so the answer must be greater than zero.

x = $\frac{-2\pm \sqrt{16}}{2}$