Problem 20 - Precalculus

Question

Evaluate the expression. \(\log _{8}(4)\)

Step-by-Step Solution

Answer

\( \log_{8}(4) = \frac{2}{3} \)

Step 1: Understand the Concept of Logarithms

The expression \( \log_{8}(4) \) asks us to find the power to which the base 8 must be raised to produce the number 4. This means finding \( x \) such that \( 8^x = 4 \).

Step 2: Convert to Exponential Equation

Convert the logarithmic expression \( \log_{8}(4) \) into its exponential form, which is \( 8^x = 4 \). Here, \( x \) is what we need to find.

Step 3: Rewrite Base in Terms of a Common Base

Express both 8 and 4 as powers of 2, since they are both powers of 2. \( 8 = 2^3 \) and \( 4 = 2^2 \). Substitute these into the equation: \((2^3)^x = 2^2\).

Step 4: Apply Exponent Rules

Apply the rule that \((a^m)^n = a^{m \times n}\) to simplify: \(2^{3x} = 2^2\).

Step 5: Equate and Solve for \( x \)

Since the bases are equal, equate the exponents: \(3x = 2\). Solve for \( x \) by dividing both sides by 3: \(x = \frac{2}{3}\).

Key Concepts

Exponential Equations

Exponential equations are equations where variables appear as exponents. These equations are widespread in mathematics because they model growth and decay and describe various natural phenomena. In our exercise, solving an exponential equation helped us find the power needed to raise one number to equal another. For example, in the equation \(8^x = 4\), finding \(x\) is the key to solving the logarithmic expression \(\log_{8}(4)\). By first understanding how an exponential equation translates from its logarithmic form, students can interpret what it means to solve for \(x\). It's crucial to remember that the essence of solving exponential equations often involves comparing powers of the same base, enabling straightforward unlocking of the variable. This showcases the interconnectedness of exponential and logarithmic functions and equations.

Exponent Rules

Exponent rules are fundamental for simplifying and solving both exponential and logarithmic equations. A crucial rule applied in this exercise was the power of a power property, which states that \((a^m)^n = a^{m\times n}\).
This allows for the simplification of complex expressions by breaking them down using multiplication of exponents, as seen when we rewrite \( (2^3)^x = 2^2 \) as \( 2^{3x} = 2^2 \). Using exponent rules not only simplifies expressions but also allows for direct comparison of equal bases, crucial in solving for unknown variables like \(x\). Understanding and applying these rules is vital in solving equations efficiently and accurately.

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m\cdot n}\)
• Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)

Mastery of these rules can transform seemingly complex expressions into manageable calculations.

Logarithmic Expressions

Logarithmic expressions tell us how many times a specific base must be multiplied by itself to achieve a certain number. For example, \(\log_{8}(4)\) seeks the exponent to which 8 is raised to yield 4.
Logarithms are the inverse operations of exponentiations – while exponentiation gives a power, logarithms help find the power given the outcome. In our exercise, we converted the logarithmic form into an exponential equation, simplifying our journey to find the solution.
Understanding logarithmic expressions relies heavily on understanding their exponential roots and interrelationships – by switching between formats, solving becomes a logical sequence.

• Logarithmic form: \(\log_b(a) = c\) means \(b^c = a\)
• Inverse nature: Logarithms undo exponentials
• Logarithms translate multiplicative processes into additive ones

Grasping these principles aids not just in solving equations, but in appreciating how these mathematical tools function in harmony.

Change of Base

The change of base formula is a very helpful tool when working with logarithms that don't easily lend themselves to common calculations. It allows us to rewrite logarithms in terms that may be more familiar or easier to compute by converting bases.
For the exercise at hand, we tackled \(\log_{8}(4)\) by using known powers of 2, turning the bases into a more workable form. The change of base formula states:

• \(\log_{b}(a) = \frac{\log_{c}(a)}{\log_{c}(b)}\)

Where \(c\) can be any valid base, often chosen to simplify calculations like base 10 or base 2. By doing this, we unlocked the problem in a more conventional setup to solve for \(x\).
Utilizing the change of base isn't just about easing calculation but also about deepening understanding of log relations and conversions. It shows the flexibility within log calculations and the simplicity that sometimes hides behind more complex structures.