Problem 10 - Algebra and Trigonometry

Question

Evaluate the expression. $$ \log _{2} 8^{33} $$

Step-by-Step Solution

Answer

99

Step 1: Apply the Power Rule of Logarithms

According to the power rule of logarithms, \( \log_b(a^c) = c \cdot \log_b(a) \). Apply this rule to our expression \( \log _{2} 8^{33} \), so it becomes \( 33 \cdot \log_2 8 \).

Step 2: Evaluate \(\log_2 8\)

We need to find the value of \(\log_2 8\). Since 8 is a power of 2 (specifically \(2^3 = 8\)), \(\log_2 8 = 3\).

Step 3: Multiply to Find the Final Answer

Now that we know \(\log_2 8 = 3\), we substitute back into our expression from Step 1: \(33 \cdot \log_2 8 = 33 \cdot 3\). Evaluate this to get \(99\).

Key Concepts

Power Rule of Logarithms

When you have a logarithmic expression where the variable inside is raised to a power, you can use the power rule of logarithms to simplify the calculation. The power rule states that \( \log_b(a^c) = c \cdot \log_b(a) \). This is extremely handy because it allows you to move the exponent in front of the logarithm, turning what might be a complex problem into a much simpler multiplication.

For example, if you need to evaluate \( \log_2 8^{33} \), you start by applying the power rule. You take the exponent 33 and bring it down in front of the log, transforming the expression into \( 33 \cdot \log_2 8 \).

Understanding and applying the power rule can make solving logarithmic equations less intimidating and more manageable. Just move the power out front, and you'll be one step closer to your solution.

Evaluating Logarithms

Evaluating a logarithm requires you to determine what power you need to raise the base to, in order to get the number inside the log expression. For example, if you have \( \log_2 8 \), you ask yourself: "2 to the power of what equals 8?"

In this case, \( 8 = 2^3 \), so \( \log_2 8 = 3 \). This means that 2 raised to the power of 3 gives you 8. The result of a logarithm is essentially the exponent. It's a way of "undoing" the exponentiation process to find out what power was used.

Practice checking numbers as powers of the given base can help in quickly evaluating logarithms. Always rewrite the number inside the log as a base raised to a specific power, if possible. This simplifies understanding and calculating the logarithm.

Properties of Exponents

Exponents are fundamental in mathematics and are entwined with logarithms. Understanding the properties of exponents underpins effectively working with logarithms.

When you have a number raised to an exponent such as \( a^b \), this tells you how many times to multiply \( a \) by itself. Key properties to remember include:

• Product of Powers: \( a^m \cdot a^n = a^{m+n} \)
• Power of a Power: \( (a^m)^n = a^{m\cdot n} \)
• Power of a Product: \( (ab)^n = a^n \cdot b^n \)

These properties help when simplifying expressions or solving equations that involve both exponents and logarithms. For example, recognizing that \( 8 = 2^3 \) provides the solution \( \log_2 8 = 3 \), as you are finding what power of the base gives the number inside the log.

By understanding these properties, you gain a solid basis for dealing with more complex expressions involving logarithms and powers.