Problem 58 - Algebra and Trigonometry

Question

Solve the logarithmic equation for \(x.\) Solve for \(x : \log _{2}\left(\log _{3} x\right)=4\)

Step-by-Step Solution

Answer

The solution is \( x = 43,046,721 \).

Step 1: Convert Logarithmic Equation to Exponential Form

Given the equation \( \log_{2}(\log_{3}x) = 4 \), we start by converting this logarithmic equation into an exponential form. The equation \( \log_{2}(A) = 4 \) can be translated into \( A = 2^{4} \). Substitute \( A \) with \( \log_{3}x \), we get \( \log_{3}x = 16 \).

Step 2: Convert Inner Logarithmic Equation to Exponential Form

Now that we have \( \log_{3}x = 16 \), we can convert this logarithmic equation into an exponential equation. The conversion of \( \log_{3}x = 16 \) translates into \( x = 3^{16} \).

Step 3: Calculate the Value of \(x\)

Using the result from the previous step, we compute \( x = 3^{16} \). By calculating, we find that \( 3^{16} = 43,046,721 \). Thus, \( x = 43,046,721 \).

Key Concepts

Exponential Form

Understanding exponential form is crucial when working with logarithms. Logarithmic and exponential functions are inverse operations, meaning one can be used to "undo" the other. A logarithmic equation like \(\log_b(y) = x\) can be rewritten in exponential form as \(y = b^x\). This conversion is a fundamental step when solving equations involving logarithms, as it allows us to express and manipulate the equation in a more straightforward form. In our exercise, we used this concept to change \(\log_{2}(\log_{3} x) = 4\) into \(\log_{3} x = 16\) by recognizing \(2^4 = 16\). This simplification is vital in breaking down the problem.

Base Conversion

Converting between different bases is another important aspect of solving logarithmic and exponential equations. The base number in a logarithmic equation indicates the number that we continually multiply to reach a given value. Sometimes, it is necessary to change or understand the base to solve an equation. In our problem, we initially dealt with a base of 2, as seen in \(\log_{2}\), and later switched to a base of 3. This conversion process involved rewriting the equation in a more approachable manner, allowing us to simplify the problem step-by-step into a more manageable form. Understanding how to navigate through base changes is key to solving complex equations efficiently.

Exponentiation

Exponentiation is the process of raising a number to the power of an exponent, a key operation when dealing with exponential equations. When we encounter a logarithmic equation rewritten as its exponential form, like \(x = 3^{16}\), we directly apply exponentiation to find the value of \(x\). Calculating \(3^{16}\) involves multiplying the number 3 by itself 16 times, which gives us the value 43,046,721. Mastery of exponentiation is crucial for solving equations that have been simplified to fit this type of mathematical expression. It allows us to grasp large values that might otherwise seem intimidating.

Mathematical Problem-Solving

Problem-solving in mathematics often involves breaking down a complex question into multiple simpler parts. It requires understanding the given equations, identifying core mathematical concepts like exponentiation, and applying strategies to find the correct solution. In this exercise, the process involved:

• Identifying and utilizing the inverse relationship between logarithms and exponentials.
• Converting between different equation forms to simplify calculations.
• Performing accurate computations to achieve a final solution.

Effective problem-solving also entails careful consideration of each step, verifying calculations, and ensuring an understanding of underlying principles. Practicing these methods can strengthen your ability to tackle diverse mathematical problems with confidence.