st3he1d0t

Answered

2022-09-04

Logarithm problem with two bases

Given

$${\mathrm{log}}_{x}9+{\mathrm{log}}_{9}x={\displaystyle \frac{10}{3}}.$$

How can I find the greatest value of x that satisfies the equation above?

Answer & Explanation

Salma Baird

Expert

2022-09-05Added 8 answers

hint: $a={\mathrm{log}}_{9}x$, then you have a quadratic in $a$:

$a+{\displaystyle \frac{1}{a}}={\displaystyle \frac{10}{3}}$

Bergsteinj0

Expert

2022-09-06Added 1 answers

Let ${\mathrm{log}}_{x}9=t;\frac{1}{{\mathrm{log}}_{x}9}=\frac{1}{t}$

Solve $t+\frac{1}{t}=\frac{10}{3}$

$3{t}^{2}+3=10t$

$3{t}^{2}-10t+3=0$

$t=\frac{10\pm \sqrt{100-\mathrm{4.3.3}}}{6}=3$ 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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