Problem 10 - Precalculus : Building Concepts and Connections

Question

Solve the exponential equation. Round to three decimal places, when needed. $$6^{x}=\frac{1}{216}$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -3\).

Step 1: Convert 216 to the base 6

216 can be written as \(6^3\), so now equation becomes \(6^{x} = \frac{1}{6^{3}}\) which simplifies to \(6^{x} = 6^{-3}\).

Step 2: Compare the exponents

Now that we have the same bases, the exponents must be equal, therefore, \(x = -3\).

Step 3: Check the answer

Substitute \(x = -3\) into the original equation to check if both sides are equal: \(6^{-3} = \frac{1}{6^{3}}\). Yes, the solution is correct.

Key Concepts

Exponent Rules

The crux of understanding exponential equations lies in grasping the exponent rules. Exponents are shorthand notation for repeated multiplication. For instance, \[a^n\] means "multiply \(a\) by itself \(n\) times."
Understanding exponent rules enables us to simplify and solve complex equations efficiently. Some pivotal rules include:

• **Product of Powers Rule**: \(a^m \cdot a^n = a^{m+n}\) - When multiplying with the same base, you add the exponents.
• **Quotient of Powers Rule**: \(\frac{a^m}{a^n} = a^{m-n}\) - When dividing with the same base, you subtract the exponents.
• **Power of a Power Rule**: \((a^m)^n = a^{m \cdot n}\) - When raising a power to another power, multiply the exponents.
• **Zero Exponent Rule**: \(a^0 = 1\) - Any base raised to the 0 power equals 1, provided \(a eq 0\).
• **Negative Exponent Rule**: \(a^{-n} = \frac{1}{a^n}\) - A negative exponent denotes a reciprocal.

In the exercise above, we used the negative exponent rule: \((6^{-3})\) is equivalent to \(\frac{1}{6^3}\). Understanding and applying these rules can greatly facilitate solving exponential equations.

Solving Equations

Solving exponential equations can seem daunting at first, but with a clear plan, it becomes manageable. The main strategy is to manipulate the equation until the exponents are isolated and the bases on both sides are identical.
Here's how you usually tackle such equations:

• **Express both sides with the same base:** If possible, rewrite the equation so both sides have the same base. This makes it easier to compare the exponents.
• **Compare the exponents:** When the bases are the same, the exponents naturally equate. This step gives a simple equation to solve for the unknown variable.
• **Check your work:** Substitute the found value back into the original equation to ensure both sides are equal. This verifies your solution is correct.

In our example, we first expressed 216 as \(6^3\). Then, by converting \(\frac{1}{6^3}\) to \(6^{-3}\), and since the bases are equal, it allowed us to equate \(x = -3\). Finally, substituting \(x = -3\) into the original equation confirmed our solution was accurate.

Change of Base

The change of base is a valuable tool in solving exponential equations, especially when direct comparison of exponents is challenging. By changing the base of a number, we can transform the equation into a more manageable form.
To apply the change of base formula:

• Choose a new base, often 10 or \(e\), to simplify computations, especially if using logs.
• Use the change of base formula:
  \[\log_b{a} = \frac{\log_c{a}}{\log_c{b}}\]
  This allows us to change from base \(b\) to base \(c\).
• Solve the equation in the new base usually involves finding the logarithm of both sides.

In our original exercise, a direct change of base wasn't required since converting both numbers to a base of 6 easily revealed the solution. However, this concept becomes exceptionally useful in more complex equations where direct conversion isn't as straightforward. Mastering the change of base rule expands your toolkit for solving a wider range of exponential problems.