Proving the Logarithmic Identity: ( log_b a cdot log_c b cdot log_a c 1 )

The logarithmic identity ( log_b a cdot log_c b cdot log_a c 1 ) can be proven in multiple ways, and the method often involves the change of base formula for logarithms. In this article, we will explore different approaches, highlighting the elegance and simplicity of each step.

Proof 1: Using the Definition of Logarithms

One of the most direct methods to prove this identity is by using the definition of logarithms. Let's start by writing the logarithms in the same base.

By definition, we know that ( b^{log_b a} a ). Taking the natural logarithm (ln) of both sides, we get:

[ log_b a frac{ln a}{ln b} ]

This can be extended to:

[ log_c b frac{ln b}{ln c} ]

and

[ log_a c frac{ln c}{ln a} ]

Multiplying these three expressions together:

[ log_b a cdot log_c b cdot log_a c frac{ln a}{ln b} cdot frac{ln b}{ln c} cdot frac{ln c}{ln a} 1 ]

The numerators and denominators cancel out, leaving us with a simple result.

Proof 2: Using the Change of Base Formula

The change of base formula for logarithms, (log_b a frac{log a}{log b}), can be used here as well. Let's substitute this into the identity:

[ log_b a cdot log_c b cdot log_a c frac{log a}{log b} cdot frac{log b}{log c} cdot frac{log c}{log a} ]

Notice how the terms cancel out:

[ frac{log a cdot log b cdot log c}{log b cdot log c cdot log a} 1 ]

This confirms the identity.

Proof 3: Using Substitution and Properties of Exponents

Another approach is to use the properties of exponents and the definition of logarithms. Let:

[ log_b a x quad text{such that} quad b^x a ]

Similarly, let:

[ log_c b y quad text{such that} quad c^y b ]

And let:

[ log_a c z quad text{such that} quad a^z c ]

Substituting these definitions into the original product:

[ log_b a cdot log_c b cdot log_a c x cdot y cdot z ]

Notice that:

[ b^x a ]

and

[ c^y b ]

Therefore:

[ a b^x c^{y cdot x} a^z cdot c^{y cdot x} ]

Since both sides equal ( c^{y cdot x} ), we have:

[ a a^{z cdot y cdot x} a^1 ]

Which simplifies to:

[ xyz 1 ]

Thus:

[ log_b a cdot log_c b cdot log_a c xyz 1 ]

Conclusion

The logarithmic identity ( log_b a cdot log_c b cdot log_a c 1 ) has been proven through multiple methods. Each approach highlights different aspects of logarithmic properties and serves as a valuable tool for understanding and applying logarithmic identities.

The key takeaways from these proofs are the utility of the change of base formula and the simplification of logarithmic expressions through substitution. These techniques are fundamental in solving complex logarithmic problems and are essential for students and professionals working in mathematics, computer science, and engineering.