Solving Simultaneous Equations with Logarithms: A Comprehensive Guide

Logarithmic equations can often require careful manipulation to solve simultaneously. This guide offers a detailed exploration of solving the given simultaneous equation utilizing logarithmic properties and calculus techniques. We will examine the conditions under which real and complex solutions do and do not exist.

Introduction

Consider the given simultaneous equations:

ln y 5 - ln x

x - ln x e-6 - 5

These equations involve natural logarithms and require careful handling to find a solution. We will first simplify each equation and then examine the conditions for real and complex solutions.

Simplifying the Equations

Starting with the first equation:

ln y 5 - ln x

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

ln y ln x 5

Combining the logarithms, we get:

ln (xy) 5

Raising both sides to the power of e gives:

xy e5

From the second equation, we have:

x - ln x e-6 - 5

Define the function:

f(x) x - ln x

The derivative of this function is:

f'(x) 1 - 1/x

Setting the derivative to zero to find the critical points:

1 - 1/x 0

Solving for x:

x 1

Evaluating f(1):

f(1) 1 - ln(1) 1

Since this value is less than the right-hand side of the second equation (which is e-6 - 5, a negative value), there are no real solutions for x that satisfy both equations simultaneously.

Proof for Real Numbers

From the simplified form of the first equation:

ln y 5 - ln x

We can express:

y e5 - ln x e5 / x

Substituting this expression into the second equation:

x - ln x e-6 - 5

This implies:

x - ln x -4.999999

Considering x > 0, both terms must be positive. However, since the right-hand side is negative, this forces:

ln x > x

This condition is impossible for real numbers because the logarithm of a number is always less than the number itself for positive values of x. Hence, there are no real solutions.

Proof for Complex Numbers

Assuming complex numbers x a bi with b ≠ 0, we substitute into the second equation:

x - ln x e-6 - 5

The complex logarithm is defined as:

ln (a bi) ln |a bi| i arg(a bi) ln |a bi| i arctan(b/a)

Substituting into the equation:

(a bi) - ln |a bi| - i arctan(b/a) -4.999999

Separating into real and imaginary parts:

1. Real part: a - ln |a bi| -4.999999

2. Imaginary part: b - arctan(b/a) 0

From the imaginary part:

b arctan(b/a)

This equation implies that:

b 0

However, this contradicts our assumption that b ≠ 0. Therefore, there are no complex solutions either.

Conclusion

In conclusion, the given simultaneous logarithmic equations have no real or complex solutions. This result is derived from the properties of logarithms, the behavior of the function in question, and the inherent constraints of complex numbers.

For a deeper understanding of logarithmic and simultaneous equations, further exploration of calculus and complex analysis is recommended.