Sure, let's work through the problem step-by-step.
We need to calculate [tex]\(2 \log_{\sqrt{2}-1} (\sqrt{2}+1)\)[/tex].
To solve this, we can rewrite the logarithm using the change of base formula. The change of base formula states that:
[tex]\[ \log_b (a) = \frac{\log_c (a)}{\log_c (b)}\][/tex]
for any positive number [tex]\(c\)[/tex].
Let's choose the natural logarithm (ln), where [tex]\(b\)[/tex] is [tex]\(\sqrt{2} - 1\)[/tex] and [tex]\(a\)[/tex] is [tex]\(\sqrt{2} + 1\)[/tex].
Using the change of base formula:
[tex]\[ \log_{\sqrt{2}-1} (\sqrt{2}+1) = \frac{\ln (\sqrt{2}+1)}{\ln (\sqrt{2}-1)}\][/tex]
Next, we need to multiply the result by 2 as given in the expression:
[tex]\[ 2 \log_{\sqrt{2}-1} (\sqrt{2}+1) = 2 \cdot \frac{\ln (\sqrt{2}+1)}{\ln (\sqrt{2}-1)}\][/tex]
From our numerical results:
- The value of [tex]\(\log_{\sqrt{2}-1} (\sqrt{2}+1)\)[/tex] calculated using natural logarithms is approximately [tex]\(-1.0000000000000002\)[/tex].
Therefore:
[tex]\[ 2 \log_{\sqrt{2}-1} (\sqrt{2}+1) = 2 \cdot (-1.0000000000000002) = -2.0000000000000004 \][/tex]
So the result of [tex]\(2 \log_{\sqrt{2}-1} (\sqrt{2}+1)\)[/tex] is approximately [tex]\(-2.0000000000000004\)[/tex].
