Problem 59 - Algebra A Combined Function

Question

If \(\log _{b} 2=0.43\) and \(\log _{b} 3=0.68\), evaluate each expression. $$ \log _{b} 8 $$

Step-by-Step Solution

Answer

\( \log_b 8 = 1.29 \)

Step 1: Rewrite Logarithm in Terms of Known Logs

The expression we have is \( \log_b 8 \). We notice that 8 can be written as a power of 2: \( 8 = 2^3 \). Thus, we can rewrite \( \log_b 8 \) using the property of logarithms which states that \( \log_b (a^n) = n \log_b a \). This gives us: \[ \log_b 8 = \log_b (2^3) = 3 \log_b 2 \].

Step 2: Substitute the Known Log Value

Now, substitute the known value of \( \log_b 2 \), which is 0.43, into the expression obtained from Step 1: \[ 3 \log_b 2 = 3 \times 0.43 \].

Step 3: Calculate the Product

Multiply the two numbers to obtain the value of \( \log_b 8 \):\[ 3 \times 0.43 = 1.29 \].

Key Concepts

Logarithm Properties

Logarithms have a set of unique properties that make them tremendously useful for solving mathematical problems. Here are some of the key properties:

• Power Rule: The logarithm of a power can be expressed as a product. Simply put, if you have a number raised to a power inside a logarithm, you can bring the power out in front as a multiplier. Mathematically, this is written as \( \log_b (a^n) = n \cdot \log_b a \). In our case, this property was used to simplify \( \log_b 8 \) by recognizing 8 as \( 2^3 \), giving us \( 3 \cdot \log_b 2 \).
• Product Rule: When multiplying two numbers together, the logarithm of their product is the sum of the logs of the numbers. It can be represented as \( \log_b (x \cdot y) = \log_b x + \log_b y \).
• Quotient Rule: Similar to the product rule, when dividing, the logarithm of a quotient is the difference of the logs. This means \( \log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y \).

These properties allow us to bend and twist logarithmic expressions into formats that are often easier to work with. In particular, the power rule was key to solving the given exercise.

Logarithmic Expressions

In mathematical problems like the one we are dealing with, it's common to transform complex expressions into simpler forms using the properties of logarithms. A logarithmic expression is essentially an equation that involves a logarithm, and its primary purpose is to express exponential relationships more conveniently.
When you face a logarithmic expression, like \( \log_b 8 \), the goal is often to break it down into known components. This is what happened in the step-by-step solution provided. We recognized that 8 is equivalent to \( 2^3 \) and used the logarithm power rule, turning \( \log_b 8 \) into \( 3 \cdot \log_b 2 \). This transformation reduced an unfamiliar logarithmic term to a product that involved a known logarithm value.
Developing an eye for identifying opportunities to apply these properties is crucial for mastering logarithmic expressions. By doing so, you'll be able to decode them into manageable pieces that make complex problems solvable.

Logarithm Calculations

Once we have rewritten a logarithmic expression into an accessible form using properties of logarithms, the next step is to perform the actual calculations. Let's delve into the calculation process using an example from our original exercise.
Having determined that \( \log_b 8 = 3 \cdot \log_b 2 \), and knowing from given data that \( \log_b 2 = 0.43 \), the task is straightforward. Simply substitute the value:

• Replace \( \log_b 2 \) in \( 3 \cdot \log_b 2 \) with \( 0.43 \).
• Calculate the product: \( 3 \cdot 0.43 = 1.29 \).

Breaking down calculations into clear steps helps in maintaining accuracy and simplifying the problem. This approach maximizes clarity, ensuring you have a clear path from the initial problem to the final solution.