Question

Show that \(\log _{a}\left(x+\sqrt{x^{2}-1}\right)+\log _{a}\left(x-\sqrt{x^{2}-1}\right)=0\)

Short answer

Using logarithmic properties and identity, we get \( \text{log}_a(1) = 0 \).

Step-by-step solution

Understand the Properties of Logarithms

Recall that \(\text{log}_a(M \times N) = \text{log}_a(M) + \text{log}_a(N)\). We are given two logarithm terms with the same base being added together.

Combine Logarithmic Terms

Using the logarithmic property from Step 1, combine the terms inside a single logarithm: \[ \text{log}_a\big[(x+\text{\text{log}_a}(x^2-1))(x-\text{\text{\text{log}_a}(x^2-1))}\big] \]

Simplify the Expression Inside the Logarithm

Simplify the expression within the logarithm: \[ (x+\text{\text{log}_a}(x^2-1))(x-\text{\text{\text{log}_a}(x^2-1))) = x^2 - (x^2 - 1) = 1 \]

Apply Logarithm of One

We now have: \[ \text{log}_a(1) \] Remember that for any base \(a\), \[ \text{log}_a(1) = 0 \]

Conclusion

Thus, \[ \text{log}_a\big(x + \text{\text{log}_a}(x^2-1)\big) + \text{log}_a\big(x - \text{\text{\text{log}_a}(x^2-1))\big) = 0 \] since \( \text{log}_a(1) = 0 \).

Key concepts

logarithmic identities

Logarithmic identities are essential tools in simplifying and solving logarithmic equations. They help to convert complex logarithmic expressions into simpler ones. Here are some common identities:

• Product Rule: \(\text{log}_a(M \times N) = \text{log}_a(M) + \text{log}_a(N)\)
• Quotient Rule: \(\text{log}_a \frac{M}{N} = \text{log}_a(M) - \text{log}_a(N)\)
• Power Rule: \(\text{log}_a(M^r) = r \times \text{log}_a(M)\)

These identities are particularly useful when dealing with the sum or difference of logarithms of the same base. In our example, \(\text{log}_a(x + \text{log}_a(x^2 - 1)) \) and \(\text{log}_a(x - \text{log}_a(x^2 - 1))\) are simplified using the Product Rule.

properties of logarithms

Understanding the fundamental properties of logarithms can greatly simplify the way we handle logarithmic equations. Here are the key properties:

• Logarithm of One: \(\text{log}_a(1) = 0\)
• Logarithm of the Base: \(\text{log}_a(a) = 1\)
• Change of Base Formula: \(\text{log}_a(M) = \frac{\text{log}_b(M)}{\text{log}_b(a)}\)

These properties help streamline complex logarithmic expressions. For instance, in the given problem, utilizing \(\text{log}_a(1) = 0\) completes the final step, confirming that the logarithmic equation is indeed zero. Each property has its specific use case, and being familiar with them is essential when simplifying logarithmic expressions.

logarithm simplification

Logarithm simplification involves breaking down complicated expressions into simpler ones, often using logarithmic identities and properties. This is useful for solving equations or making the expressions more manageable.
In our example, \(\text{log}_a\big[(x+\text{\text{log}_a}(x^2-1))(x-\text{\text{\text{log}_a}(x^2-1))}\big]\) utilizes both the Product Rule and properties of mathematical operations.
The intermediate step involves recognizing that \((x + \text{\text{log}_a}(x^2 - 1))(x - \text{\text{log}_a}(x^2 - 1)) = x^2 - (x^2 - 1) = 1\). Therefore, simplifying to \(\text{log}_a(1) \), which equals 0, is crucial. This process illustrates how combining properties and identities can reduce a complex equation to its simplest form. The concept of logarithm simplification is fundamental in many areas of mathematics, including solving exponential and logarithmic equations.