Problem 60 - A Graphical Approach to College Algebra

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 is \( x = 4 \).

Step 1: Rewrite the Logarithmic Equation

Start by setting the original equation aside: \( \log_{2}(\log_{2} x) = 1 \). Recognize that what you need to do is to simplify the equation. Here, treat the inner log expression \( \log_{2} x \) as your variable, let's call it \( y \). So the outer log equation becomes \( \log_{2} y = 1 \).

Step 2: Solve for the Inner Logarithm

The equation \( \log_{2} y = 1 \) is saying that \( y \) is the power to which the base 2 must be raised to get the result of 1. To solve, rewrite this in exponential form: \( y = 2^{1} \). Thus, \( y = 2 \).

Step 3: Substitute Back to Find x

Since \( y = 2 \) corresponds to \( \log_{2} x = y \), substitute back to find \( x \): \( \log_{2} x = 2 \). This is another equation to solve, where \( x \) is the power to which the base 2 must be raised to give 2. In exponential form, it's \( x = 2^{2} \).

Step 4: Calculate and Express the Exact Solution

Now, solve for \( x \) using the exponential equation: \( x = 2^{2} = 4 \). Therefore, the value of \( x \) that satisfies the original equation is \( x = 4 \).

Step 5: Verify Solution With a Calculator

Use a calculator if needed to verify: Calculate \( \log_{2} 4 \) and confirm that it equals 2. Then, calculate \( \log_{2} 2 \) to confirm that it gives 1 as expected. This confirms that \( x = 4 \) is indeed the correct solution.

Key Concepts

exponential functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. These functions are central in many growth processes in nature and are foundational to understanding logarithmic relationships. The general form of an exponential function is given by \( f(x) = a^{x} \), where \( a \) is a positive constant. In this form, \( x \) represents the exponent or power to which the base \( a \) is raised.

One of the essential characteristics of exponential functions is their rapid rate of growth or decay, depending on whether the base is greater than or less than one. This makes them crucial in fields such as finance, biology, and physics.

In our solution, we see exponential functions in action when we rewrite the logarithmic equation. By changing \( \log_{2} y = 1 \) into an exponential form, we solve \( y = 2^{1} \), which simplifies to \( y = 2 \). This translation between logarithmic and exponential forms is a vital skill in solving such problems effectively.

logarithms

Logarithms are the inverse operations of exponential functions. If exponential functions involve raising a base to an exponent to get a number, then logarithms solve the reverse question: given a number, to what exponent must a specific base be raised to produce that number?

The general logarithmic form is \( \log_{b} a = c \), which translates to the exponential form \( b^{c} = a \). Here, \( b \) is the base of the logarithm, \( a \) is the result, and \( c \) is the exponent.

Logarithms can simplify complex multiplication problems into addition problems, which is particularly useful in higher-level mathematics and science. In solving the exercise, we utilize logarithmic properties by recognizing that \( \log_{2} \) is consistent through the equations, enabling simplification to a solvable form. This step underscores the importance of understanding these properties in manipulating and solving mathematical equations effectively.

problem solving

Problem-solving in the context of logarithmic and exponential equations involves understanding and applying key mathematical principles. The goal is to simplify the given problem step-by-step until reaching a solution.

Here are some tips for effective problem solving in this context:

• Break down the problem: Simplifying complex logarithmic expressions into manageable steps aids in the comprehension and simplification of the problem.
• Translate the language: Use the relationship between logarithms and exponentials to convert the equation formats. This translation often reveals simpler equations that are easier to solve.
• Verify your results: Always check the solutions with the original problem to ensure correctness and accuracy. This can often be done using a calculator, making sure the logical steps hold true throughout.

By following these strategies, students can develop a systematic approach to tackling and solving logarithmic equations, enhancing their mathematical proficiency and confidence.