# Solving 3log(x)=6-2x: A Beginner's Guide

• crazylum
In summary, the conversation discusses the difficulty of solving the equation 3log(x)=6-2x and suggests using a numerical approach like Newton's method or MATLAB's solver to find an approximate solution. It is also mentioned that the equation can be rewritten using the Lambert W-function for an exact solution, but for practical purposes, bisection and Newton's method are recommended.
crazylum
Where does one begin to solve the equation 3log(x)=6-2x?

The solution to that equation is not expressible in terms of elementary functions.

Looks like material for Lambert's W (or Newton's method).

you could use a numerical approach like Newton's method to find a solution. if I am not mistaken you can say that x>0 since ln(x) is undefined otherwise and then rewrite your equation as $x^{3}e^{2x}-e^{6}=0$

an approximate solution is possible through pt of intersection of the 2 graphs...but i don't think there's any way to find an exact solution save hit n trial

MATLAB's solver gave:

$$\frac{\mathrm{e}^{2}}{\mathrm{e}^{\omega\!\left(\ln\!\left(\frac{2}{3}\right) + 2\right)}}$$

Where $$\omega$$ is given by: http://en.wikipedia.org/wiki/Wright_Omega_function.

Good call by whoever said it needed the Lambert W-function ^_^

But for practical purposes, just use bisection+Newton.

## 1. What does "3log(x)" mean in this equation?

The term "3log(x)" represents the logarithm of x raised to the power of 3. In other words, it is the exponent or power to which the base 10 must be raised to get the value of x cubed.

## 2. How do I solve this equation?

To solve this equation, start by isolating the logarithmic term on one side of the equation. In this case, you can add 2x to both sides, which will result in 3log(x) + 2x = 6. Next, use the power property of logarithms to rewrite the left side of the equation as log(x^3). Then, you can use the definition of logarithms to rewrite the equation as x^3 = 10^(6-2x). Finally, take the cube root of both sides to isolate x and solve for its value.

## 3. Why is it important to solve logarithmic equations?

Logarithmic equations are useful in many scientific fields, particularly in chemistry, physics, and biology. They can help us solve problems involving exponential growth and decay, as well as problems involving orders of magnitude. Additionally, they are used in data analysis to transform non-linear relationships into linear ones, making them easier to interpret and analyze.

## 4. Are there any restrictions on the values of x in this equation?

Yes, there are restrictions on the values of x in this equation. Since logarithms are only defined for positive numbers, x must be greater than 0. Additionally, in order for the equation to be solvable, the value inside the logarithm (x) must be positive. This means that x cannot be equal to or less than 0.

## 5. What are some real-life applications of logarithmic equations?

Logarithmic equations have numerous real-life applications. They are used in finance to calculate compound interest, in earthquake magnitude scales to measure the intensity of earthquakes, and in pH levels to measure the acidity or basicity of a substance. They are also used in signal processing, seismology, and many other fields.

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