Problem 9 - Precalculus Functions and Graphs

Question

Determine the number that can be used in place of the question mark to make the equation true. $$2^{?}=64$$

Step-by-Step Solution

Answer

The number is 6.

Step 1: Understand the Equation

The given equation is \(2^? = 64\). We need to determine the value of the exponent that makes this equation true.

Step 2: Rewrite the Equation Using Logarithms

To solve for the unknown exponent, we can use logarithms. Taking the logarithm base 2 on both sides of the equation gives us: \( \log_2(2^?) = \log_2(64) \).

Step 3: Apply the Logarithm Power Rule

Using the power rule of logarithms, \( \log_b(a^c) = c \log_b(a) \), we can simplify the left side to: \(? = \log_2(64)\).

Step 4: Calculate the Logarithm

We need to find the power to which 2 must be raised to get 64. We know that \(64 = 2^6\). Therefore, \(\log_2(64) = 6\).

Step 5: Verify the Solution

To verify, substitute 6 back into the original equation: \(2^6 = 64\). This confirms that our solution is correct.

Key Concepts

logarithms

Logarithms are a critical tool in solving exponential equations. Essentially, a logarithm helps to determine the power or exponent that one number must be raised to produce another number. For instance, in the equation \(2^{?} = 64\), the question mark represents the exponent we need to find.

To rewrite this using logarithms, we use the fact that the logarithm of a number is the exponent to which the base must be raised to retrieve that number. In our example, we take the base 2 logarithm of both sides: \( \log_2(2^{?}) = \log_2(64) \). The left side can now be simplified using the power rule, which significantly simplifies our task.

By taking the logarithm, a complicated exponent problem becomes a much simpler linear problem. In summary, logarithms allow us to 'undo' the exponentiation process and solve for the unknown exponent.

power rule of logarithms

The power rule of logarithms is a powerful property that allows us to deal with exponents more easily. This rule states that \( \log_b(a^c) = c \log_b(a) \). In the context of our current problem, it is used to simplify \( \log_2(2^{?}) \) to \( ? \cdot \log_2(2) \).

Let's break it down:

• When you have a logarithm of a power, you can move the exponent in front of the logarithm.
• This transforms a more complex expression into a linear one, making it easier to solve.

For our equation \( \log_2(2^{?}) = \log_2(64) \), the power rule converts the left side into \( ? \cdot \log_2(2) \). Because \( \log_2(2) \) equals 1, the expression simplifies to \(? = \log_2(64)\).

In short, the power rule of logarithms is an essential technique that simplifies the process of solving for unknown exponents in exponential equations.

solving exponential equations

Solving exponential equations often involves using logarithms to isolate the exponent. Let's walk through the process, step-by-step.

First, we recognize the equation we need to solve: \(2^{?}=64\). To determine the value of the exponent, we take the logarithm of both sides of the equation using the same base as the exponent. Here, we take the base 2 logarithm: \( \log_2(2^?) = \log_2(64) \). This step is crucial because it allows us to use the power rule of logarithms.

Next, by applying the power rule \( \log_b(a^c) = c \log_b(a) \), the equation simplifies to \( ? \cdot \log_2(2) = \log_2(64) \). Since \( \log_2(2) \) equals 1, this further simplifies to \(? = \log_2(64)\).

To find \( \log_2(64) \), we determine the power of 2 that equals 64. Knowing that \(64 = 2^6\), it follows that \( \log_2(64) = 6\), giving us \(? = 6\).

Finally, we can verify our solution by substituting 6 back into the original equation: \(2^6 = 64\), confirming that our answer is indeed correct. In conclusion, using logarithms to solve exponential equations simplifies the problem and leads directly to the solution.