Key to "To solve the given equation for x. Given..."

High School
Algebra
Pre-Algebra
Factors and multiples

cistG

Answered question

2021-07-31

To solve the given equation for x.
Given equation:

\log_{5} (x^{2} - 5 x + 6) - \log_{5} (x - 2) = 1

Answer & Explanation

hajavaF

Skilled2021-08-01Added 90 answers

Concept used:
Quotient property of logarithm: $\log_{a} b - \log_{a} c = \log_{a} (\frac{b}{c})$
$\log_{a} a = 1$ .
Calculation:
$\log_{5} (\frac{x^{2} - 5 x + 6}{x - 2}) = 1$
Next step is no factor the numerator $x^{2} - 5 x + 6$ .
To factorize the above trinomial, first step is to find the two multiples of the constant term 6 so that their addition will result the coefficient of x which is -5.
So, $6 = (- 3) \cdot (2)$
Addition of -3 and 2 will result -5.
So, the factors form of:
$x^{2} - 5 x + 6 = (x - 3) (x - 2)$
Hence, the equation will be:
$\log_{5} (\frac{(x - 3) (x - 2)}{x - 2}) = 1$
$\log_{5} (x - 3) = 1$

Cancel out $(x - 2)$ from both numerator and denominator.
$\log_{5} (x - 3) = \log_{5} 5$

Since, $\log_{a} a = 1$ .
$x - 3 = 5$
$x - 3 + 3 = 5 + 3$

By adding 3 from each sides of the equation.
$x = 8$ .

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