Question

$$ \text { Solve: } \log _{2}\left[\log _{4}\left(\log _{3} x\right)\right]=0 $$

Short answer

The solution is \( x = 81 \).

Step-by-step solution

Understand the Problem

The given problem is \ \ \log _{2}\left[\log _{4}\left(\log _{3} x\right)\right]=0. Our goal is to find the value of \(x\).

Solve the Outer Logarithm

Start by solving the outer logarithm equation \ \ \log _{2}\left[\log _{4}\left(\log _{3} x\right)\right]=0. Since \(\log_{2}(y) = 0\) when \(y = 1\), we get: \ \ \ \log_{4}(\log_{3} x) = 1.

Evaluate the Second Logarithm

Next, we solve \(\log_{4}(\log_{3} x) = 1\). Since \(\log_{4}(y) = 1\) when \(y = 4\), we find that: \ \ \log_{3} x = 4.

Solve the Inner Logarithm

Now solve for \(x\) in \(\log_{3} x = 4\). This implies: \ \ \ x = 3^{4} = 81.

Verify the Solution

Double-check each step to ensure the solution satisfies the original equation. Substituting \(x = 81\), verify: \ \ \ \log_{3}(81) = 4, \ \log_{4}(4) = 1, \ \log_{2}(1) = 0.

Key concepts

Logarithmic Equations

Logarithmic equations involve unknowns inside logarithms. You solve them by rewriting the equation to isolate the variable or by using logarithmic properties.
To solve \(\text{log}_2\left[\text{log}_4\left(\text{log}_3 x\right)\right] = 0\), we broke it down step by step:
1. Understand that the equation is \(\text{log}_2(\text{some expression}) = 0\).
2. Knowing that a logarithm of 0 represents 1, rewrite the expression as \[\log_4(\log_3(x)) = 1\].
3. Translate the equation into a more straightforward form, one layer at a time.
Another example is solving \(\text{log}_b(y) = c\) by converting it to an exponential form: \(y = b^c\). Using these techniques allows you to manipulate logarithms effectively.

Logarithmic Properties

Logarithmic properties simplify complex problems and allow solving nested logarithms. Key properties include:
1. **Product Property:** \(\text{log}_b(mn) = \text{log}_b(m) + \text{log}_b(n)\)
2. **Quotient Property:** \(\text{log}_b\left(\frac{m}{n}\right) = \text{log}_b(m) - \text{log}_b(n)\)
3. **Power Property:** \(\text{log}_b(m^p) = p \cdot \text{log}_b(m)\)
4. **Change of Base Formula:** \(\text{log}_b(a) = \frac{\text{log}_k(a)}{\text{log}_k(b)}\)
Identifying and applying these properties can simplify problems significantly. In our exercise, we used the property that \(\text{log}_a(a) = 1\) to convert \(\text{log}_4(4)\) and solve for the inner nested logarithm.

Change of Base Formula

The change of base formula allows converting a logarithm to a different base, making it easier to solve or simplify. The formula is:
\[\text{log}_b(x) = \frac{\text{log}_k(x)}{\text{log}_k(b)}\]
This is particularly useful when dealing with multiple logarithms of different bases.
For example, to solve \(\text{log}_2(16)\), if you are more comfortable with base 10, you can convert it using the change of base formula:
\[\text{log}_2(16) = \frac{\text{log}_{10}(16)}{\text{log}_{10}(2)}\].
Applying this to our nested logarithm problem, while it wasn't directly necessary, knowing how to use the change of base formula equips you with a powerful tool for other logarithmic problems. Practice using this formula alongside basic properties to master logarithmic equations.