Problem 88 - College Algebra Essentials

Question

Let \(\log _{b} 2=A\) and \(\log _{b} 3=C .\) Write each expression in terms of \(A\) and \(C\). \(\log _{b} \sqrt{\frac{3}{16}}\)

Step-by-Step Solution

Answer

The expression \(\log _{b} \sqrt{\frac{3}{16}}\) is equal to \(0.5[C - 4A]\)

Step 1: Identify the given expressions

Note that \(\log _{b} 2 = A\) and \(\log _{b} 3 = C\). The task is to express \(\log _{b} \sqrt{\frac{3}{16}}\) using \(A\) and \(C\).

Step 2: Apply the square root in the expression

Remember that the square root operation can be written as the exponent of \(0.5\). So, the expression \(\log _{b} \sqrt{\frac{3}{16}}\) can be rewritten as \(\log _{b} ({\frac{3}{16}})^{0.5}\).

Step 3: Apply Logarithmic Quotient Rule

The Quotient Rule of logarithms states that the logarithm of a quotient is equal to the difference of the logarithms. Rewriting the expression using this, we get: \(0.5 \cdot \log _{b}{\frac{3}{16}} = 0.5 * (\log _{b}3 - \log _{b}16)\).

Step 4: Apply Logarithmic Power Rule

Remember that \(\log _{b}16\) can be written as \(4\log _{b}2\) using the power rule of logarithms. Substituting this, the expression becomes: \(0.5 * (\log _{b}3 - 4*\log _{b}2)\).

Step 5: Expressing in terms of A and C

Replace \(\log _{b}2\) with \(A\) and \(\log _{b}3\) with \(C\) to get the final expression: \(0.5[C - 4A]\).

Key Concepts

Logarithmic Quotient Rule

When we work with logarithms, sometimes we encounter expressions in the form of a quotient, or a division. In such cases, we have a useful tool called the Logarithmic Quotient Rule. This rule helps us simplify logarithmic expressions involving a quotient.

According to the Quotient Rule:

• The logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator.

Mathematically, if you have \[\log_b\left(\frac{M}{N}\right)\]It can be expressed as:\[\log_b M - \log_b N\]This means, to find the logarithm of a fraction, you subtract the logarithm of the denominator from the logarithm of the numerator. This rule is extremely helpful when simplifying complex expressions and helps to break them down into manageable parts.

In our original exercise, this rule also allows us to express complex logarithmic equations in terms of known variables like \(A\) and \(C\).

Logarithmic Power Rule

The Logarithmic Power Rule is another valuable tool in our logarithm toolbox. This rule simplifies expressions where a number inside a logarithm is raised to a power.

Here's how the Power Rule works:

• It states that the logarithm of a number raised to an exponent can be rewritten by bringing the exponent in front of the logarithm as a multiplier.

In mathematical terms, for any number \(M\) raised to the power of \(n\),\[\log_b (M^n) = n \cdot \log_b M\]This neat trick of bringing down the power as a multiplier can considerably simplify the calculations and reorganize the expression neatly.

In our step-by-step solution, this rule was applied to rewrite \(\log_b{16}\) as \(4 \cdot \log_b{2}\). This makes the expression simpler and easier to manage, especially when substituting known values for variables.

Change of Base Formula

The Change of Base Formula in logarithms allows us to change the base of a logarithm to another base, which can be more convenient in certain calculations. Often, natural logarithm (base \(e\)) or common logarithm (base \(10\)) are vastly preferred due to the availability of calculators and computational tools that support these bases.

The Change of Base Formula is:

• For any logarithm \(\log_b M\), it can be rewritten using a new base \(k\) as:\[\frac{\log_k M}{\log_k b}\]

This formula is incredibly useful when evaluating logarithms with uncommon bases, as it allows conversion into a base compatible with the most accessible computational tools.

While it was not applied directly in the original exercise, understanding the Change of Base Formula is essential for expanding your logarithm skills and solving a broader range of problems with efficiency.