Problem 54 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The solution is \(x = 3^{16} = 43,046,721\).

Step 1: Understand the Problem

The goal is to find the value of \(x\) such that \(\log_2(\log_3 x) = 4\). We will solve this by working backwards through the logarithms.

Step 2: Eliminate the Outer Logarithm

To eliminate the outer logarithm \(\log_2(\cdot)\), rewrite the equation as an exponential to get \(\log_3 x = 2^4 = 16\).

Step 3: Eliminate the Inner Logarithm

Now eliminate the inner logarithm by rewriting \(\log_3 x = 16\) as an exponential. This gives \(x = 3^{16}\).

Step 4: Calculate \(3^{16}\)

Compute \(3^{16}\), which equals 43,046,721. This is done through repeated multiplication or using a calculator.

Key Concepts

Exponential Functions

Exponential functions are all about repeatedly multiplying a base number by itself. They follow the form: \( f(x) = a^x \), where \( a \) is the base, and \( x \) is the exponent. Exponents are powerful because they grow numbers very quickly.

Exponential functions are the flip side of logarithms in mathematics. This relationship means if you have an equation with a logarithm, like \( \log_b(y) = x \), it can be expressed in its exponential form as \( y = b^x \). For example, if you know that \( \log_2(y) = 4 \), then you can convert it using the base of 2 in the form of an exponential equation: \( y = 2^4 = 16 \).

Understanding exponential functions will help you solve logarithmic equations, as shown in the problem where it goes from \( \log_2(\log_3 x) = 4 \) to \( \log_3 x = 16 \) after applying the exponential conversion. Once you know that, you can proceed with solving for \( x \) using similar principles.

Change of Base Formula

The change of base formula is like a handy translator for converting one logarithm base into another. This is very useful if you are using a calculator that only has a specific base, like the natural logarithm (ln) or base 10 log.

The formula is: \( \log_b(x) = \frac{\log_k(x)}{\log_k(b)} \), where \( b \) is the original base and \( k \) is the new base. It allows you to switch the base of the logarithm to whatever suits you better, whether it's base 10, base \( e \), or something else.

In this exercise, although we didn't explicitly use the change of base formula, knowing how it works makes you flexible. It shows you how every logarithmic function is part of a bigger system. You gain more control over solving problems by effectively choosing which base to work in, potentially simplifying your calculations.

Simplifying Logarithms

Simplifying logarithms means making a complex logarithmic expression easier to understand or work with. This often involves using properties of logarithms such as product, quotient, and power rules.

Some basic rules include:

• \( \log_b(xy) = \log_b(x) + \log_b(y) \)
• \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \)
• \( \log_b(x^k) = k \cdot \log_b(x) \)

These rules allow you to take difficult expressions and break them into smaller, more manageable parts.

In the given problem, while we didn't explicitly simplify using these rules, simplifying the expression \( \log_3 x = 16 \) directly to solve for \( x \), uses an understanding of the relationship between exponents and logarithms which can be seen as a form of simplification. Understanding these principles can help when dealing with more complex equations involving logarithms.