The value of log base2 log base 3 log base 3 19683

The value of log base2 log base 3 log base 3 19683

The Correct Answer and Explanation is:

We are asked to evaluate the expression: log⁡2(log⁡3(log⁡3(19683)))\log_2(\log_3(\log_3(19683)))log2​(log3​(log3​(19683)))

Step 1: Evaluate log⁡3(19683)\log_3(19683)log3​(19683)

We first determine the value of log⁡3(19683)\log_3(19683)log3​(19683).
Recall that: 39=196833^9 = 1968339=19683

So: log⁡3(19683)=9\log_3(19683) = 9log3​(19683)=9

Step 2: Evaluate log⁡3(9)\log_3(9)log3​(9)

Now substitute the previous result: log⁡3(log⁡3(19683))=log⁡3(9)\log_3(\log_3(19683)) = \log_3(9)log3​(log3​(19683))=log3​(9)

Since: 32=93^2 = 932=9

We have: log⁡3(9)=2\log_3(9) = 2log3​(9)=2

Step 3: Evaluate log⁡2(2)\log_2(2)log2​(2)

Now evaluate: log⁡2(log⁡3(log⁡3(19683)))=log⁡2(2)\log_2(\log_3(\log_3(19683))) = \log_2(2)log2​(log3​(log3​(19683)))=log2​(2)

And since: log⁡2(2)=1\log_2(2) = 1log2​(2)=1

Final Answer:

1\boxed{1}1​

Explanation

To solve the expression log⁡2(log⁡3(log⁡3(19683)))\log_2(\log_3(\log_3(19683)))log2​(log3​(log3​(19683))), we proceed step by step from the innermost logarithm outward. This approach ensures clarity and accuracy by resolving the smallest nested part first.

The innermost part is log⁡3(19683)\log_3(19683)log3​(19683). To simplify this, we observe whether 19683 is a power of 3. By evaluating powers of 3, we find that 39=196833^9 = 1968339=19683. Therefore, log⁡3(19683)=9\log_3(19683) = 9log3​(19683)=9 because logarithms ask the question “To what power must the base be raised to get the argument?” In this case, 39=196833^9 = 1968339=19683, so the answer is 9.

Next, we move to the next layer: log⁡3(9)\log_3(9)log3​(9). We now substitute the result of the previous step, giving us log⁡3(9)\log_3(9)log3​(9). Since 32=93^2 = 932=9, it follows that log⁡3(9)=2\log_3(9) = 2log3​(9)=2.

Finally, we evaluate log⁡2(2)\log_2(2)log2​(2). This comes from replacing the previous result in the outermost logarithm: log⁡2(2)\log_2(2)log2​(2). Because any logarithm of a number with the same base equals 1 — that is, log⁡b(b)=1\log_b(b) = 1logb​(b)=1 — we conclude that log⁡2(2)=1\log_2(2) = 1log2​(2)=1.

This step-by-step breakdown highlights the importance of understanding both exponentiation and logarithmic rules. Each step relies on basic logarithmic identities and properties of exponents. By simplifying from the innermost part outward, we avoid confusion and ensure logical consistency throughout the process. The final answer is 1\boxed{1}1​.