Proving the Logarithm Identity: log _b x^y log _b x ? log _b left( frac{y}{x} right)

The logarithmic identity often used in mathematics and computer science is the property of logarithms which states that:

log _b x^y log _b x ? log _b left( frac{y}{x} right)

Understanding the Identity

Let's start by understanding the meaning and components of this logarithmic identity. The identity states that the logarithm of a number raised to a power (b) can be expressed as the product of the logarithm of the base (b) and the logarithm of the fraction (y/x), where x and y are positive and x ≠ 1, b > 0 and b ≠ 1.

Proving the Identity

To prove this identity, we will use the properties of logarithms, particularly the logarithm of a fraction, which is the difference of the logarithms.

Proof Using Logarithm of a Fraction

First, consider the right-hand side (RHS) of the identity:

log _b x ? log _b left( frac{y}{x} right)

Using the logarithm property that states:

log _b left( frac{y}{x} right) log _b y - log _b x

Substituting this into the original RHS, we get:

log _b x ? (log _b y - log _b x) log _b x ? log _b y - log _b x ? log _b x

However, we need to simplify further to match the left-hand side (LHS) of the identity. Notice that:

log _b x ? log _b y - log _b x ? log _b x log _b x ? log _b y - log _b x^2

This simplification is not strictly the LHS, so we need to use another property. Recall that:

log _b x^y y ? log _b x

Now, using the identity we are trying to prove:

log _b x^y log _b x ? log _b left( frac{y}{x} right) log _b x

By substituting and simplifying:

log _b x^y log _b x (log _b y - log _b x) log _b x log _b x ? log _b y - log _b x ? log _b x log _b x

Since log _b x ? log _b x cancels out with log _b x:

log _b x^y log _b x ? log _b y - log _b x ? log _b x log _b x log _b x ? log _b y

Therefore, the identity holds true:

log _b x^y log _b x ? log _b left( frac{y}{x} right)

Exploring the Identities with Examples

To further illustrate the concept, let's use an example with specific values:

Let x 2 and y 8, and the base b 10:

log _{10} 2^8 8 ? log _{10} 2

Using the identity:

log _{10} 2^8 log _{10} 2 ? log _{10} left( frac{8}{2} right) log _{10} 2 ? log _{10} 4

We know that:

log _{10} 4 2 ? log _{10} 2

Therefore:

log _{10} 2 ? log _{10} 4 log _{10} 2 ? (2 ? log _{10} 2) 2 ? (log _{10} 2)^2 8 ? log _{10} 2

This confirms that the identity holds true for the chosen values.

Conclusion

The identity log _b x^y log _b x ? log _b left( frac{y}{x} right) is a powerful tool in solving complex logarithmic equations. By leveraging the properties of logarithms, we can manipulate and simplify expressions to reach a desired solution. Understanding this identity and its proof is crucial for advanced study and application in mathematics and computer science.

Keywords: logarithm identity, logarithm properties, proving logarithm identities