Is there an analytic solution for the equation $\log_{2}{x}+\log_{3}{x}+\log_{4}{x}=1$?

Question

I am looking for a close form solution for below equation.

$$\log_{2}{x}+\log_{3}{x}+\log_{4}{x}=1.$$

I solve it by graphing, but I don't know is there a way to find $x$ analytically ?

Answer 1

It's $$\ln{x}\left(\frac{1}{\ln2}+\frac{1}{\ln3}+\frac{1}{\ln4}\right)=1$$ or $$\ln{x}=\frac{1}{\frac{1}{\ln2}+\frac{1}{\ln3}+\frac{1}{\ln4}}$$ or $$x=e^{\frac{1}{\frac{1}{\ln2}+\frac{1}{\ln3}+\frac{1}{\ln4}}}$$

Answer 2

Hint: $$\log[b](x) = \dfrac{\ln(x)}{\ln(b)}$$

Answer 3

Let $y = log_{10}x$

Then

$y(\frac{1}{log2}+\frac{1}{log3}+\frac{1}{log4}) = 1$

Find y

and then $x = 10^y$

Answer 4

$$\tag1 \log_{2}{x}+\log_{3}{x}+\log_{4}{x}=1$$

$$\tag2 \frac{\log{x}}{\log{2}}+\frac{\log{x}}{\log{3}}+\frac{\log{x}}{\log{4}} = 1$$

$$\tag3 \left(\frac{1}{\log{2}}+\frac{1}{\log{3}}+\frac{1}{\log{4}}\right)\log{x} = 1$$

$$\tag4 \left(\frac{1}{\log{2}}+\frac{1}{\log{3}}+\frac{1}{\log{4}}\right) = \frac{1}{\log{x}}$$

$$\tag5 \left(\frac{1}{\log{3}} + \frac{3}{\log{4}}\right) = (\log{x})^{-1}$$

$$\tag6 \left(\frac{1}{\log{3}} + \frac{3}{\log{4}}\right)^{-1} = \log{x}$$

$$\tag7 10^{\left(\frac{1}{\log{3}} + \frac{3}{\log{4}}\right)^{-1}}= x$$

$$10^{\frac{1}{\frac{1}{\log{3}} + \frac{3}{\log{4}}}}= x$$

Answer 5

$$ log_v {u}= \dfrac{log_a u}{log_a v} $$

valid for any arbitrary base $a$, need not necessarily be $10$ or $e$.

So

$$\log_a{x}\left(\dfrac{1}{\log_a2}+\dfrac{1}{\log_a3}+\frac{1}{\log_a4}\right)=1$$

or

$$x=a^\left({\dfrac{1}{\dfrac{1}{\log_a2}+\dfrac{1}{\log_a3}+\dfrac{1}{\log_a4}}}\right)=f(a)$$

EDIT1:

apparently but in fact $ \ne any \,f(a)$

Consequently $x$ is indeterminate... as we are free to choose any base.

Sorry, since $ a^{log_a u} = \,u $ it has a cancelling effect ... hastily over sighted on pure visual/structural inspection of the above. ( I wished only to say "consequently $x$ is independent of base... as we are free to choose any base " ) etc. ...because I placed more value on base independence as relevant for this question.

g[u_] = Log[u] (1/Log[3] + 1/Log[4] + 1/Log[2]);
ParametricPlot[{a, a^(1/g(a))}, {a, 1, 12}, GridLines -> Automatic]

that yields a base free $ x \approx 1.3844.$

Answer 6

Just to give a slightly different approach, let $x=e^u$. Then

$$1=\log_2x+\log_3x+\log_4x=u(\log_2e+\log_3e+\log_4e)$$

so

$$x=e^u=e^{1/(\log_2e+\log_3e+\log_4e)}$$

Note: $e$ here can be any positive number, not just $2.718281828\ldots$. For example, we might write

$$x=4^{1/(\log_24+\log_34+\log_44)}=4^{1/(3+\log_34)}$$

Since $3\lt4\lt9$, we have $1\lt\log_34\lt2$, with the value of $\log_34$ closer to $1$ than it is to $2$. If we use the crude approximation $\log_34\approx1$, we get

$$x\approx4^{1/4}=\sqrt2\approx1.414$$

Since $\log_34$ is actually bigger than $1$, the exact value of $x$ is somewhat less than $\sqrt2$. Closer calculation gives

$$x\approx4^{1/4.2618595}\approx4^{0.234639}\approx1.384416$$