Problem 60 - A Graphical Approach to Precalculus with Limits

Question

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log _{2}\left(\log _{2} x\right)=1$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 4 \).

Step 1: Understand the Given Equation

The given equation is \( \log_2(\log_2 x) = 1 \). This means we need to find a value of \( x \) such that the logarithm of \( x \) to base 2, when again taken as a logarithm to base 2, equals 1.

Step 2: Simplify the Inside Equation

To simplify \( \log_2(\log_2 x) = 1 \), we rewrite it using the properties of logarithms. If \( \log_b(y) = z \), then \( y = b^z \). Here, \( \log_2 x = 2^1 = 2 \).

Step 3: Solve for x

Now solve \( \log_2 x = 2 \). Using the same property of logarithms, we infer that \( x = 2^2 = 4 \).

Step 4: Verify Your Solution

Substitute \( x = 4 \) back into the original equation to verify: \( \log_2(\log_2 4) = \log_2(2) = 1 \), which matches the given equation. This confirms the solution is correct.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverse functions of exponential functions. If you have an exponential function, such as \( y = b^x \), its corresponding logarithmic form is \( x = \log_b(y) \). This relationship is fundamental in understanding how logarithms work.
Logarithms answer the question: "To what power must the base, \( b \), be raised to produce a given number, \( y \)?" For example, \( \log_2(8) \) asks what power you need to raise 2 to get 8, and the answer is 3, since \( 2^3 = 8 \).
When dealing with logarithmic functions, it's important to identify both the base of the logarithm and the number you're taking the logarithm of, as this determines the output of the function. Remember that the base \( b \) must always be a positive number other than 1.

Properties of Logarithms

The properties of logarithms can simplify complex logarithmic expressions and equations. Understanding these properties will make it easier to solve logarithmic equations efficiently. Some key properties include:

• Product Property: \( \log_b(mn) = \log_b(m) + \log_b(n) \)
• Quotient Property: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \)
• Power Property: \( \log_b(m^n) = n\cdot\log_b(m) \)
• Change of Base Formula: \( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \), which allows you to change the base of a logarithm.

These properties allow you to break down logarithmic expressions just as you would with basic arithmetic operations. For example, if you're working on the equation \( \log_2(x) = 2 \), you can apply the power property in reverse to deduce \( x = 2^2 \).

Solving Logarithmic Equations

Solving logarithmic equations involves using the properties of logarithms to isolate the variable. Let's look at our original example, \( \log_2(\log_2 x) = 1 \). We first need to simplify the inner logarithm. This is achieved by rewriting the equation using exponential form.
1. Start by setting the inner logarithm to equal 2, since \( \log_2(2) = 1 \). This step uses the knowledge that \( \log_b(y) = z \) implies \( y = b^z \).2. Therefore, \( \log_2 x = 2 \).3. Now, apply the same logic to solve for \( x \): \( x = 2^2 = 4 \).
Finally, it's always a good practice to check your solution by substituting back into the original equation. Verifying confirms that the solution is indeed correct—an essential step to ensure no errors were made in calculations.