Logarithmic Equation solve log_(3x)3+log_(x/3)3=5/12

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In summary, the conversation discusses solving a logarithmic equation with two different bases. The suggestion is to change both bases to the same value, and an example is given using base 3x. The final equation is simplified by letting y equal one of the logarithms, and the solution for x is found by solving for y and plugging it back into the original logarithm.
  • #1
Yankel
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Dear all,

I wish to solve the following logarithmic equation:

\[log_{3x}3+log_{\frac{x}{3}}3=\frac{5}{12}\]

My intuition was to start with changing the base of both logarithms to 10 (or any other number), but couldn't continue from there. Can you assist please ? Is there a meaning to the fact that both bases involves 3 in them ?

Thank you in advance.
 
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  • #2
Yes, you certainly want the two logarithms to the same base but there is nothing special about base 10. Since one logarithm is already to base "3x" I would be inclined to change the other to base 3x also. If [tex]y= log_{x/3}(3)[/tex] then [tex]3= (x/3)^y= x^y/3^y[/tex]. So [tex]x^y= 3^{y+1}[/tex] and then [tex](3x)^y= 3^yx^y= 3^{2y+ 1}[/tex]. Taking the logarithm, base 3x, of both sides, [tex]y= log_{3x}(3^{2y+1})= (2y+1)log_{3x}(3)[/tex]. Solving that for y, [tex](1- 2log_{3x}(3))y= log_{3x}(3)[/tex] so [tex]y= log_{x/3}(3)= \frac{log_{3x}(3)}{1- 2log_{3x}(3)}[/tex]

The original equation, [tex]log_{3x}(3)+ log_{x/3}(3)= \frac{5}{12}[/tex] becomes [tex]log_{3x}(3)\left(1+ \frac{log_{3x}(3)}{1-2log_{3x}(3)}\right)= \frac{5}{12}[/tex].

To simplify, let [tex]y= log_{3x}(3)[/tex] so the equation is [tex]y\left(1+ \frac{y}{1- 2y}\right)= \frac{5}{12}[/tex]. Solve that for y then solve [tex]log_{3x}(3)= y[/tex] for x.
 

FAQ: Logarithmic Equation solve log_(3x)3+log_(x/3)3=5/12

1. What is a logarithmic equation?

A logarithmic equation is an equation that contains a logarithm, which is the inverse function of an exponential function. It is written in the form logb(x) = y, where b is the base, x is the input, and y is the output.

2. How do I solve a logarithmic equation?

To solve a logarithmic equation, you need to isolate the logarithm on one side of the equation. This can be done by using the properties of logarithms, such as the product, quotient, and power rules. Once the logarithm is isolated, you can raise both sides of the equation to the same power to eliminate the logarithm and solve for the variable.

3. What is the base of a logarithm?

The base of a logarithm is the number that is raised to a power in order to get a certain value. For example, in the equation log2(8) = 3, the base is 2 because 23 = 8.

4. Can a logarithmic equation have more than one solution?

Yes, a logarithmic equation can have more than one solution. However, it is important to check for extraneous solutions, which are solutions that do not satisfy the original equation.

5. How do I solve the equation log3x(3) + logx/3(3) = 5/12?

To solve this equation, you can use the product rule of logarithms to combine the two logarithms into one. This will result in logx(9) = 5/12. Then, you can raise both sides of the equation to the power of x to eliminate the logarithm. This will result in x5/12 = 9. Finally, you can solve for x by taking the 12th root of both sides, which will give you the value of x.

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