Question

How do you solve \[{{3}^{{{x}^{2}}-1}}=729\]?

Answer 1

Hint: To solve this question, we should know some of the logarithmic properties. We will use the property that \[\log {{a}^{b}}=b\log a\]. Also, for a general quadratic equation \[a{{x}^{2}}+bx+c=0\], we can find the roots of the equation using the formula method as \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]. By substituting the value of coefficient, in the above formula we can find the roots.

Complete step by step solution:
We are given to solve the equation \[{{3}^{{{x}^{2}}-1}}=729\]. We know that 729 is 3 raised to 6. Hence, we can express the given equation as \[{{3}^{{{x}^{2}}-1}}={{3}^{6}}\]. Taking logarithm on both sides of this equation, we get \[\log {{3}^{{{x}^{2}}-1}}=\log {{3}^{6}}\]. We know the logarithmic property \[\log {{a}^{b}}=b\log a\], using this we can express the equation as
\[\Rightarrow \left( {{x}^{2}}-1 \right)\log 3=6\log 3\]
Cancelling out common factor from both sides of the above equation, we get
\[\Rightarrow {{x}^{2}}-1=6\]
Simplifying the above equation, we get
\[\Rightarrow {{x}^{2}}-7=0\]
The above equation is a quadratic. Comparing the standard form of a quadratic equation \[a{{x}^{2}}+bx+c=0\] we get \[a=1,b=0\And c=-7\]. We can find the roots of the equation using formula method as \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\], substituting the values of coefficients we get
\[\begin{align}
  & \Rightarrow x=\dfrac{-0\pm \sqrt{{{0}^{2}}-4(1)(-7)}}{2(1)} \\
 & \Rightarrow x=\dfrac{\pm 2\sqrt{7}}{2} \\
 & \therefore x=\pm \sqrt{7} \\
\end{align}\]
Thus, the roots of the equation are \[x=\sqrt{7}\] or \[x=-\sqrt{7}\].

Note: We can check if the solution is correct or not by substituting the values in the equation. Substitute \[x=\sqrt{7}\] in the equation, we get LHS as \[{{3}^{{{\left( \sqrt{7} \right)}^{2}}-1}}={{3}^{7-1}}={{3}^{6}}=729\], as RHS is 729. As LHS and RHS are the same, the solution is correct. Now, substitute \[x=-\sqrt{7}\] in the equation, we get LHS as \[{{3}^{{{\left( -\sqrt{7} \right)}^{2}}-1}}={{3}^{7-1}}={{3}^{6}}=729\], as RHS is also 729. As LHS and RHS are the same this solution is also correct.