Enter logarithmic equation expression in this form \log(x^2 – 5) – \log x = \log 4
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
\log(x^2 – 5) – \log x = \log 4can be rewritten using the properties of logarithms as:
\frac{x^2-5}{x}=\log4This can be further simplified to:
(x^2 – 5)/x = 4Multiplying both sides by x to get rid of the fraction:
x^2 – 5 = 4xRearranging 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{-b\pm \sqrt{b^2-4ac}}{2a} 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.
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