Problem 58 - Precalculus Mathematics for Calculus

Question

Solve for \(x: \) $$\log _{2}\left(\log _{3} x\right)=4$$

Step-by-Step Solution

Answer

\(x = 43046721\)

Step 1: Identify the Inner Logarithm

The problem involves nested logarithms. Start by identifying the inner logarithm, which is \(\log_{3} x\). The outer logarithm is \(\log_{2}\).

Step 2: Express the Constraint from the Outer Logarithm

Since \(\log_{2}(\log_{3} x) = 4\), rewrite this equation in exponential form: \(\log_{3} x = 2^{4}\). Calculate \(2^{4}\) to get 16.

Step 3: Resolve the Inner Logarithm

Now, solve \(\log_{3} x = 16\). This expression can be rewritten in its exponential form: \(3^{16} = x\).

Step 4: Calculate the Value

Finally, calculate \(3^{16}\). This requires calculating the power of 3 raised to 16, yielding \(x = 43046721\).

Key Concepts

nested logarithms

Nested logarithms are expressions where one logarithm is inside another. This can look a bit tricky at first, but let's break it down. When you see something like \( \log_{b}( \log_{c}(x) ) \), it just means you have two layers of logarithms. The inner logarithm, in this case \( \log_{c}(x) \), is the first one we need to deal with. After that, we consider the outer logarithm \( \log_{b} \).To solve problems with nested logarithms, it’s crucial to work from the inside out. Here’s how:

• Identify the Inner Logarithm: Locate the most deeply nested logarithm and start by solving or simplifying it first.
• Simplify Layer by Layer: Once the innermost part is simplified, move to the outer logarithm.

Working step by step makes these problems easier and more straightforward.

exponential form

Transforming a logarithmic equation into exponential form is a powerful technique for solving equations. It involves using the definition of logarithms: if \( \log_{b}(a) = c \), then it's equivalent to saying \( b^{c} = a \). This method is especially useful when dealing with nested logarithms, as it allows us to simplify and solve the equations in a stepwise manner.### How It's Done

• Rewrite the Equation: Convert the logarithmic form to exponential form. For example, if we have \( \log_{2}(y) = 4 \), it means \( 2^{4} = y \).
• Calculate the Power: Once in exponential form, calculate the power to find the value of the expression.

Applying this approach turns a nested logarithm problem into a more manageable format that can be easily tackled.

solving for x

Solving for \( x \) involves finding the value of \( x \) that makes the equation true. With logarithmic equations, this often requires multiple steps, especially if nested logarithms are involved.### The Resolution Process

• Isolate the Logarithmic Expression: Start by focusing on the innermost logarithmic expression, simplifying or rewriting it into exponential form.
• Solve the Resulting Equation: Work with the simplified equation to calculate and solve for \( x \).

In our example, we need to deal with the result of the outer and then the inner logarithm to find \( x \). Once the expression \( \log_{2}(\log_{3}x) \) is simplified to \( 16 \), you solve \( \log_{3}x = 16 \) by recognizing \( x = 3^{16} \). Calculating this gives the final value for \( x \). This step-by-step approach ensures you don't miss any hidden details.