Problem 85 - Algebra and Trigonometry

Question

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

\(\log _{b} 8 = 3A\)

Step 1: Rewrite the number 8 in terms of 2

The number 8 can be written in terms of 2 as \(2^3\). Hence, \(\log _{b} 8\) can be written as \(\log _{b} 2^3\)

Step 2: Apply the power rule of logarithms

The power rule of logarithms states that \(log_b(m^n) = n \cdot log_b(m)\), hence \(\log _{b} 2^3\) can be written as \(3 \cdot \log _{b} 2\) or \(3A\) since we are given that \( \log _{b} 2 = A\)

Step 3: Final expression

So, from the above steps, we can express \(\log _{b} 8\) in terms of A and C as \(3A\). Thus, \(\log _{b} 8 = 3A\)

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a fundamental property that simplifies the process of working with logarithmic expressions. Essentially, it helps in breaking down expressions with exponents into a more manageable form.

This rule states: if you have a logarithm of a number raised to an exponent, \(\log_b(m^n) = n \cdot \log_b(m)\), you're allowed to "bring down" the exponent as a multiplier in front of the logarithm.

So, in the context of the problem, the expression \(\log_b(2^3)\) can be rewritten by using the power rule as \(3 \cdot \log_b(2)\). By substituting the given value of \(\log_b(2)\) which is \(A \), we have \(3A \).

• This makes the expression simpler and more straightforward.
• It's a powerful tool to use when dealing with logarithmic equations.

Logarithmic Expressions

Logarithmic expressions may initially appear complex but break down into simpler parts by leveraging logarithmic identities and rules. At their core, these expressions are based on the idea of identifying what power a base must be raised to in order to achieve a certain value.

For example, in the exercise, we start with \(\log_b 8\). To work with this expression using given information, we need to rewrite it in terms of \(A\) and \(C\).
- Recognizing 8 as \(2^3\) is integral as it connects back to \(\log_b 2\), or \(A\).- Through the power rule, you can then manipulate and simplify \(\log_b 8\) to become \(3A\).Logarithmic expressions may involve complex transformations, but by understanding the rules, they can be broken down easily.

Logarithmic Identities

Logarithmic identities are crucially helpful when dealing with logarithms, offering shortcuts and methods for simplification that make complicated expressions more manageable. Knowing these identities allows you to transform and simplify expressions for easier computation.

Some common logarithmic identities include:

• Product Rule: \(\log_b(x \cdot y) = \log_b(x) + \log_b(y)\)
• Quotient Rule: \(\log_b(x/y) = \log_b(x) - \log_b(y)\)
• Change of Base Formula: \(\log_b(x) = \frac{\log_c(x)}{\log_c(b)}\)

In our exercise, the use of the power rule (an identity itself) was pivotal to rewrite the expression in terms of already known variables; there's no direct need for the other identities, but awareness of them aids in broader problem-solving contexts.