Problem 85 - College Algebra

Question

In Exercises \(83-88,\) let \(\log _{b} 2=A\) and \(\log _{b} 3=\) C. Write each expression in terms of \(A\) and \(C\). $$ \log _{b} 8 $$

Step-by-Step Solution

Answer

After completing the steps, the final answer to express \(\log _{b} 8\) in terms of \(A\) and \(C\) is \(3A\).

Step 1: Breakdown of the Problem

In the given exercise, we are tasked to rewrite \(\log _{b} 8\) in terms of \(A\) and \(C\). One important thing to remember is that 8 can be expressed as a power of 2, specifically \(2^3\)

Step 2: Apply logarithmic identities

Now that we've identified how 8 can be written in terms of 2, we can use one of the logarithmic identities, specifically the power rule. The power rule states that \(\log _{b} m^n = n \log _{b} m\). Thus, applying the rule gives: \(\log _{b} 8 = \log_{b} 2^3 = 3 \log_{b} 2\)

Step 3: Final Interpretation

The final step is to simply replace \(\log _{b} 2\), which is given as \(A\) in the problem. Therefore, \(\log _{b} 8 = 3A\).

Key Concepts

Logarithmic Identities

Logarithmic identities are essential rules that help us manipulate and understand logarithms better. They act like tools for simplifying complex logarithmic expressions. Let's have a quick review of the most common identities:

• Product Rule: \( \log_b (mn) = \log_b m + \log_b n \)
• Quotient Rule: \( \log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n \)
• Power Rule: \( \log_b (m^n) = n \log_b m \)

These identities allow you to transform a logarithm in various ways. For instance, in our exercise, the power rule was used to express \( \log_b 8 \) in terms of \( A \) directly. Understanding these identities can greatly simplify working with logarithms, especially in decomposing or reconstructing numbers and equations.

Change of Base Formula

The change of base formula is a handy tool when dealing with logarithms of different bases. Sometimes, you encounter logarithms with a base that isn't very convenient for calculation, like a calculator that only provides logarithms base 10 or natural logarithms. The formula is:

• \( \log_b a = \frac{\log_k a}{\log_k b} \)

Here, \( k \) is a new base of your choice, often 10 or \( e \) for natural logs.
This formula was not needed in the given exercise since the focus was on expressing the logarithm in terms of known values \( A \) and \( C \). Nevertheless, it's an efficient method for calculations when you're switching between different logarithmic bases.

Exponents

Exponents are fundamental in mathematics since they describe repeated multiplication, similar to how logarithms describe repeated division. When you see something like \( 2^3 \), it means 2 multiplied by itself 3 times: \( 2 \times 2 \times 2 = 8 \).

In the given problem, recognizing that 8 can be expressed as \( 2^3 \) was pivotal. This understanding allowed us to use the power rule of logarithms effectively. Remember, when you see a number, think about whether it can be broken down into powers of known bases. This break down technique simplifies working with logarithms and exponents in solving mathematical problems.

Power Rule

The power rule of logarithms is a powerful and straightforward tool for simplifying logarithmic expressions involving exponents. The rule states that \( \log_b (m^n) = n \log_b m \). This means you can "bring down" the exponent as a multiplier in front of the logarithm, making it easier to work with.

In our solution, the power rule is what allowed us to transform \( \log_b 2^3 \) into \( 3 \log_b 2 \). This conversion turned a complex-looking logarithm into a simpler expression that could easily be translated into terms of \( A \). Understanding and applying this rule can help you unlock other scenarios where exponents are present inside logarithmic evaluations.