Indices, surds and logarithms equation

Question

Can we use indices or using logarithms is better ?

$$\dfrac{5^x-5^{-x}}{2} = 3$$ Solve $x$ correct to $4$ decimal places.

Answer 1

We want to solve the equation $5^x-5^{-x}=6$. Let $y=5^x$. Then our equation becomes $$y-\frac{1}{y}=6.$$ Multiply though by $y$. We get the Quadratic Equation $y^2-6y-1=0$. Now can you continue? You can certainly find $y$ to very good accuracy (one of the roots will be no good). After that, logs will be helpful.

Remark: Or else one can just apply a numerical method like the Newton Method directly to the original equation.

If we just want the answer, many calculators have a "Solve" button that will do the job. Or else one can ask Wolfram Alpha for the answer. There are many other programs that will solve equations numerically to high accuracy.