Solving Simultaneous Equations Involving Logarithms

Simultaneous equations can often involve logarithmic terms, which might seem challenging at first. However, with the proper steps and understanding of logarithmic properties, these equations can be solved effectively. In this article, we will solve the following simultaneous equations:

(2x^{ln 2} 3y^{ln 3}) (3^{ln x} 2^{ln y})

Let's approach this step by step, ensuring that we manipulate the logarithms correctly to isolate variables.

Step-by-Step Solution Guide

Step 1: Simplify the First Equation

The first equation can be rewritten using logarithmic properties. Let's start with:

(2x^{ln 2} 3y^{ln 3})

Applying the logarithm on both sides, we get:

(ln(2x^{ln 2}) ln(3y^{ln 3}))

Using the properties of logarithms: (ln(ab) ln a ln b) and (ln(a^b) bln a), we can simplify it further:

(ln 2 ln x^{ln 2} ln 3 ln y^{ln 3})

Which simplifies to:

(ln 2 (ln 2) ln x ln 3 (ln 3) ln y)

Let’s rewrite this as:

(ln 2^2 (ln 2)ln x ln 3^2 (ln 3)ln y)

Step 2: Simplify the Second Equation

The second equation is:

(3^{ln x} 2^{ln y})

Applying the natural logarithm on both sides, we get:

(ln(3^{ln x}) ln(2^{ln y}))

Using the properties of logarithms, we can simplify this to:

(ln x cdot ln 3 ln y cdot ln 2)

Which can be rearranged to:

(frac{ln x}{ln y} frac{ln 2}{ln 3})

Let:

(k frac{ln 2}{ln 3})

Then:

(ln x k ln y)

Step 3: Substitute and Solve

Substituting (ln x k ln y) into the first equation, we get:

(ln 2^2 (ln 2)k ln y ln 3^2 (ln 3)ln y)

Rearranging gives:

(ln 2^2 - ln 3^2 (k ln 2 - ln 3)ln y 0)

So:

((k ln 2 - ln 3)ln y ln 3^2 - ln 2^2)

Therefore:

(ln y frac{ln 3^2 - ln 2^2}{k ln 2 - ln 3})

Step 4: Solve for (ln y)

Substitute the value of (k frac{ln 2}{ln 3}) in the equation:

(ln y frac{ln 3^2 - ln 2^2}{frac{ln 2}{ln 3} ln 2 - ln 3})

Simplify further:

(ln y frac{2ln 3 - 2ln 2}{frac{ln 2^2}{ln 3} - ln 3})

(ln y frac{2ln 3 - 2ln 2}{frac{2ln 2}{ln 3} - ln 3})

Step 5: Solve for (ln x)

Using (ln x k ln y):

(ln x frac{ln 2}{ln 3} cdot frac{2ln 3 - 2ln 2}{frac{2ln 2}{ln 3} - ln 3})

Step 6: Calculate Explicit Values for (x) and (y)

To find the explicit values of (x) and (y), we need to:

Calculate (ln 2) and (ln 3) Substitute these values into the equations Exponentiate to find (x) and (y)

For example, if (ln 2 0.6931) and (ln 3 1.0986), we can compute:

(k frac{0.6931}{1.0986} 0.6309)

(ln y frac{2cdot 1.0986 - 2cdot 0.6931}{0.6309cdot 2cdot 0.6931 - 1.0986})

(ln y frac{2.1972 - 1.3862}{0.8859 - 1.0986})

(ln y frac{0.811}{-0.2127})

(ln y -3.806)

Therefore:

(y e^{-3.806} approx 0.0217)

Similarly, (ln x 0.6309 cdot -3.806 approx -2.402)

(x e^{-2.402} approx 0.0907)

Conclusion

This approach will lead to the values of (x) and (y). You can calculate these numerically to find the specific values if needed. It's crucial to be careful and methodical when dealing with logarithms, as most people may not fully understand how to manipulate them effectively.

Additionally, it's worth noting that there might be other superior methods to solve such equations. Different approaches can lead to different insights and solutions, so it's always beneficial to explore multiple methods.