Question

Show that \(\log _{a}(\sqrt{x}+\sqrt{x-1})+\log _{a}(\sqrt{x}-\sqrt{x-1})=0\)

Short answer

The given equation is valid because \ \ \log_{a}(\sqrt{x} + \sqrt{x-1}) + \log_{a}(\sqrt{x} - \sqrt{x-1}) = \log_{a}(1) = 0.

Step-by-step solution

Use Logarithm Properties

Start by using the property of logarithms: \ \ \ \text{\[\log_b (m) + \log_b (n) = \log_b (m \cdot n)\]} \ \ The equation can be written as: \ \[\log_{a}((\sqrt{x}+\sqrt{x-1}) \cdot (\sqrt{x}-\sqrt{x-1})) = 0.\]

Simplify the Argument

Simplify the product inside the logarithm using the difference of squares formula: \ \[(\sqrt{x} + \sqrt{x-1})(\sqrt{x} - \sqrt{x-1}) = x - (x-1) = 1.\] \ The equation becomes: \ \[\log_{a}(1) = 0.\]

Use Logarithm Property of 1

Recall that for any base \ a \, \ \[\log_{a}(1) = 0.\] \ Therefore, the equation is valid and simplifies to: \ \[0 = 0.\]

Key concepts

Logarithmic Identities

Logarithmic identities are fundamental tools that make solving logarithmic equations easier. One of the essential identities is the product property, which states: \[\log_b (m) + \log_b (n) = \log_b (m \cdot n)\]
This property is derived from the definition of logarithms and helps in transforming log addition into log multiplication. This way, complex logarithmic expressions can become simpler and more manageable.
In our exercise, we used this property to combine two logarithms: \[\log_{a}(\sqrt{x}+\sqrt{x-1}) + \log_{a}(\sqrt{x}-\sqrt{x-1}) = \log_{a}((\sqrt{x}+\sqrt{x-1})(\sqrt{x}-\sqrt{x-1}))\]
By transforming the addition into a single logarithm, we can simplify our work.

Difference of Squares

The difference of squares formula is a powerful algebraic tool: \[a^2 - b^2 = (a+b)(a-b)\]
This formula states that the product of the sum and difference of the same two terms equals the difference of the squares of those terms. By recognizing patterns that fit this structure, we can simplify many algebraic expressions.
In our problem, we applied this formula to the expression within the logarithm: \[\sqrt{x}^2 - \sqrt{x-1}^2 = x - (x-1) = 1\]
Thus, we saw that: \[ (\sqrt{x} + \sqrt{x-1})(\sqrt{x} - \sqrt{x-1}) = 1\]
Using the difference of squares helped us reduce the expression inside the logarithm to a much simpler form.

Properties of Logarithms

Understanding properties of logarithms is essential for solving logarithmic problems effectively. Some key properties include:

• The product property: \(\log_b(mn) = \log_b(m) + \log_b(n)\)
• The quotient property: \(\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\)
• The power property: \(\log_b(m^n) = n \log_b(m)\)

Additionally, the property of logarithms involving 1 is particularly important: \[\log_a(1) = 0\]
This property tells us that any number raised to the power of zero equals one. Applying this in our exercise: \[\log_a(1) = 0\]
This confirms our simplified logarithmic equation, ensuring it holds true. Mastery of these properties allows for efficient manipulation and simplification of logarithmic expressions.