Problem 27 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the exponential equation. $$\left(\frac{1}{8}\right)^{x}=64$$

Step-by-Step Solution

Answer

The solution to the equation \(\left(\frac{1}{8}\right)^{x}=64\) is \(x = -2\).

Step 1: Rewrite both sides of the equation with base 2

Write \(\left(\frac{1}{8}\right) = 2^{-3}\) and \(64 = 2^{6}\). This leads us to \(2^{-3x} = 2^{6}\).

Step 2: Equate exponents

Since the bases are equal (\(2\) in this case), it implies that the exponents are also equal. This leads to \(-3x = 6\).

Step 3: Solve for x

Solve for \(x\), which gives you \(x = \frac{-6}{-3} = -2\).

Key Concepts

Base Conversion

When dealing with exponential equations, it is often crucial to work with similar or the same bases. This involves converting the given numbers into expressions with the common base. In our exercise, we began with the equation \(\left(\frac{1}{8}\right)^{x}=64\). Both these numbers can be written as powers of 2:

• The fraction \(\frac{1}{8}\) converts to \(2^{-3}\) because \(8 = 2^3\) and \(\frac{1}{8} = 8^{-1}\), which simplifies to \(2^{-3}\).
• The number 64 is equal to \(2^6\) since \(64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2\).

Converting numbers this way allows you to compare exponents directly, a strategy often needed in more advanced exponential problems. It's crucial to choose a base that simplifies the equation, often the smallest prime base common between both numbers.

Equating Exponents

Once the bases of the two sides of the equation are the same, we can move to the next step: equating the exponents. In the equation \(2^{-3x} = 2^{6}\), the bases are already equal, which directly allows us to set their exponents equal to each other.

This simplifies the equation to solving \(-3x = 6\). This principle is founded on the property of exponential functions, which states that if \(a^m = a^n\), then it must be the case that \(m = n\).

Therefore, solving the exponent gives us the value for \(x\). Keep in mind that this ability to equate exponents only works when the bases are identical, which is why base conversion is such an important preliminary step.

Solving for x

After equating the exponents in our exponential equation, the next task is solving for \(x\). From \(-3x = 6\), solving for \(x\) involves isolating it on one side of the equation. We can do this by dividing both sides by \(-3\).

• First, divide both sides of the equation by \(-3\): \(-3x = 6\) becomes \(x = \frac{6}{-3}\).
• After simplifying, this gives \(x = -2\).

This process requires using basic algebraic principles, where you perform equal operations on both sides of an equation to maintain its balance. Solving for \(x\) in such cases not only tests your algebra skills but also reinforces your understanding of exponential equations and their properties.