ln52ln4+log(x2+8)51=log5(7x2+9x)
x=?
\( \frac{2\ln4}{\ln5}+\frac{1}{\log_{(x^2+8)}5}=\log_5(7x^2+9x) \)
\( x=\text{?} \)
\( \frac{2\log_78}{\log_74}+\frac{1}{\log_43}\times\log_29= \)
\( \frac{\log_311}{\log_34}+\frac{1}{\ln3}\cdot2\log3= \)
\( \frac{\log_76-\log_71.5}{3\log_72}\cdot\frac{1}{\log_{\sqrt{8}}2}= \)
\( -3(\frac{\ln4}{\ln5}-\log_57+\frac{1}{\log_65})= \)
To solve the given equation, follow these steps:
We start with the expression:
Use the change-of-base formula to rewrite everything in terms of natural logarithms:
Multiplying the entire equation by to eliminate the denominators:
By properties of logarithms (namely the product and power laws), combine the left side using the addition property:
Since the natural logarithm function is one-to-one, equate the arguments:
Rearrange this into a standard form of a quadratic equation:
Attempt to solve this quadratic equation using the quadratic formula:
Where , , and .
Calculate the discriminant:
The discriminant is positive, suggesting real solutions should exist, however, verification against the domain constraints of logarithms (arguments must be positive) is needed.
After solving , the following is noted:
The polynomial does not yield any values in domains valid for the original logarithmic arguments.
Cross-verify the potential solutions against original conditions:
Solutions obtained do not satisfy these together within the purview of the rational roots and ultimately render no real value for .
Therefore, the solution to the problem is: There is no solution.
No solution
\( \frac{1}{\ln4}\cdot\frac{1}{\log_810}= \)
\( \frac{\log_8x^3}{\log_8x^{1.5}}+\frac{1}{\log_{49}x}\times\log_7x^5= \)
\( \frac{1}{\log_x3}\times x^2\log_{\frac{1}{x}}27+4x+6=0 \)
\( x=\text{?} \)
\( \frac{1}{\log_{2x}6}\times\log_236=\frac{\log_5(x+5)}{\log_52} \)
\( x=\text{?} \)
Find X
\( \frac{1}{\log_{x^4}2}\times x\log_x16+4x^2=7x+2 \)
Find X