Exponential And Logarithmic Functions

To solve the problem, we need to prove that using logarithmic identities.

1. Define the Initial Expression:
Let . This represents a logarithmic relationship between the variables.

2. Take the Logarithm of Both Sides:
Taking the logarithm (base ) on both sides:

.

3. Apply the Change of Base Formula:
Using the logarithmic identity , we convert both sides to natural logarithms (or base-10):

.

4. Rearrange the Equation:
Cross-multiply to isolate :

.

5. Rewrite in Logarithmic Form:
This implies , which means .

6. Substitute Back the Original Expression:
Since , we substitute to get:

.

7. Prove the Given Identity:
By symmetry, .

Therefore: .

Final Answer:
The proof shows that .