Question

find value of x or solve for x​

Answer 1

$\textsf{\large{\underline{Solution}:}}$

We have to solve the logarithmic equation.

$\rm \longrightarrow \log( {x}^{2} - 21 ) = 2$

When no base is given, base is assumed to be 10. Therefore:

$\rm \longrightarrow \log_{10} ( {x}^{2} - 21 ) = 2$

By definition of logarithm:

$\rm \longrightarrow {x}^{2} - 21= {10}^{2}$

$\rm \longrightarrow {x}^{2} - 21 - {10}^{2} = 0$

$\rm \longrightarrow {x}^{2} - 21 -100 = 0$

$\rm \longrightarrow {x}^{2} - 121 = 0$

$\rm \longrightarrow {x}^{2} - {11}^{2} = 0$

$\rm \longrightarrow (x + 11)(x - 11) = 0$

$\rm \longrightarrow x = \pm11$

★ So, the values of x satisfying the given equation is 11 and -11.

$\textsf{\large{\underline{Learn More}:}}$

$\rm 1. \: \: {a}^{n} = b \implies log_{a}(b) = n$

$\rm 2. \: \: log_{a}(1) = 0, \: a \neq0,1$

$\rm 3. \: \: log_{a}(a) = 1, \: a \neq0,1$

$\rm 4. \: \: log_{a}(x) = log_{a}(y) \implies x = y$

$\rm 5. \: \: log_{e}(x) = ln(x)$

$\rm6. \: \: log_{a}(x) + log_{a}(y) = log_{a}(xy)$

$\rm7. \: \: log_{a}(x) - log_{a}(y) = log_{a} \bigg( \dfrac{x}{y} \bigg)$

$\rm 8. \: \: log_{a}( {x}^{n} ) = n\log_{a}(x)$

$\rm 9. \: \: log_{a}(m) = \dfrac{ log_{b}(m) }{ log_{b}(a) },m > 0,b > 0,a \ne1,b \ne1$

$\rm 10. \: \: log_{a}(b) = \dfrac{1}{ log_{b}(a) }$