st3he1d0t

2022-09-04

Logarithm problem with two bases
Given
${\mathrm{log}}_{x}9+{\mathrm{log}}_{9}x=\frac{10}{3}.$
How can I find the greatest value of x that satisfies the equation above?

Do you have a similar question?

Salma Baird

Expert

hint: $a={\mathrm{log}}_{9}x$, then you have a quadratic in $a$:
$a+\frac{1}{a}=\frac{10}{3}$

Still Have Questions?

Bergsteinj0

Expert

Let ${\mathrm{log}}_{x}9=t;\frac{1}{{\mathrm{log}}_{x}9}=\frac{1}{t}$
Solve $t+\frac{1}{t}=\frac{10}{3}$
$3{t}^{2}+3=10t$
$3{t}^{2}-10t+3=0$
$t=\frac{10±\sqrt{100-4.3.3}}{6}=3$ or $\frac{1}{3}$.
So ${\mathrm{log}}_{x}9=3$ or $\frac{1}{3}$
Solve them, you will not get a neat answer, but an answer.

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