Logarithmic Equation

A logarithmic equation is an equation that involves logarithms. Logarithmic equations are solved by using properties of logarithms or by converting the equation into exponential form.

The expression

can be rewritten using the properties of logarithms as:

Solution

This can be further simplified to:

Multiplying both sides by x to get rid of the fraction:

Rearranging the equation into a quadratic form:
x^2 – 4x – 5 = 0

Now we can solve this quadratic equation by factoring or using the quadratic formula. Let’s use the quadratic formula:

x=\frac{-(-4)\pm \sqrt{(-4)^2-4*1*(-5)}}{2*1}
x = \frac{ (4 ± \sqrt{(16 – (-20)} } {2}
x = \frac{  (4 ± √36)} {2}
x = \frac{ (4 ± 6)} {2}

Therefore, the solutions to the equation are:

x = (4 + 6)/2 = 10/2 = 5
x = (4 – 6)/2 = -2/2 = -1

So, the values of x that satisfy the original logarithmic equation are x = 5 and x = -1.