Chapter 5: Q 101. (page 309)

Show that $\log_{a} (x + \sqrt{x^{2} - 1}) + \log_{a} (x - \sqrt{x^{2} - 1}) = 0$ .

Short Answer

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The result is shown in the explanation.

Step by step solution

Step 1. Given information.

Consider the given information.

$\log_{a} (x + \sqrt{x^{2} - 1}) + \log_{a} (x - \sqrt{x^{2} - 1}) = 0$

Step 2. Explanation.

Consider the left side of the given equation.

$\log_{a} (x + \sqrt{x^{2} - 1}) + \log_{a} (x - \sqrt{x^{2} - 1})$

Step 3. Calculation.

Apply the logarithm rule to combine the left side.

$\begin{array}{rcl} \log_{a} (x + \sqrt{x^{2} - 1}) + \log_{a} (x - \sqrt{x^{2} - 1}) & = & \log_{a} (x + \sqrt{x^{2} - 1}) (x - \sqrt{x^{2} - 1}) \\ = & \log_{a} (x^{2} - (x^{2} - 1)) \\ = & \log_{a} (1) \\ = & 0 \end{array}$

The given statement is true as the left side is equal to the right side.

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Show that $\log_{a} (x + \sqrt{x^{2} - 1}) + \log_{a} (x - \sqrt{x^{2} - 1}) = 0$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Explanation.

Step 3. Calculation.

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Show thatlogax+x2-1+logax-x2-1=0.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Explanation.

Step 3. Calculation.

One App. One Place for Learning.

Most popular questions from this chapter

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Show that $\log_{a} (x + \sqrt{x^{2} - 1}) + \log_{a} (x - \sqrt{x^{2} - 1}) = 0$ .