Alg9.4.17

Find the value of x if $\log_5(x)+\log_x(5)=\frac{5}{2}\,$

Let $\log_5(x)=a\,$

Hence

$\log_x(5)=\frac{1}{a}\,$

Substituting these two values in the given equation,it looks as

$a+\frac{1}{a}=\frac{5}{2}\,$

Simplifying the equation we get

$\frac{a^2+1}{a}=\frac{5}{2}\,$

Cross multiplying the two fractions we get

$2a^2-5a+2=0\,$

Solving this quadratic equation

$(2a-1)(a-2)=0\,$

Hence the values of a are 2 and 1/2

Now substituting the value of a

we get

$\log_5(x)=2\,$

$x=25\,$

Second value,

$\log_5(x)=\frac{1}{2}\,$

$x=\sqrt{5}\,$