Problem 54 - Precalculus Mathematics for Calculus

Question

Use the Laws of Logarithms to combine the expression. $$\log _{a} b+c \log _{a} d-r \log _{a} s$$

Step-by-Step Solution

Answer

\( \\log_{a} \left( \\frac{b \\cdot d^c}{s^r} \right) \\.

Step 1: Apply the Product Rule of Logarithms

The Product Rule of Logarithms states that \( \log_{a} m + \log_{a} n = \log_{a} (m \cdot n) \.\) In our expression, \( \log_{a} b \) and \( \log_{a} d \) can be combined: \[ \log_{a} b + c \log_{a} d = \log_{a} (b \cdot d^c) \.\] Here, \(d^c\) results from the power inside the logarithm.

Step 2: Apply the Power Rule of Logarithms

Rewrite the expression \( \log_{a} d^c \) using the Power Rule: \(c \log_{a} d \) translates to \(\log_{a} (d^c) \). This is already incorporated in Step 1, resulting in \(\log_{a}(b \cdot d^c) \.\) There are no further power terms to handle.

Step 3: Apply the Quotient Rule of Logarithms

The Quotient Rule of Logarithms states that \( \log_{a} m - \log_{a} n = \log_{a} \left( \frac{m}{n} \right) \.\) Apply this rule to combine \( \log_{a} (b \cdot d^c)\) and \( r \log_{a} s \,\) rewritten as \( \log_{a} (s^r):\)\[ \log_{a} \left( \frac{b \cdot d^c}{s^r} \right) \.\]

Key Concepts

Quotient Rule of Logarithms

The Quotient Rule of Logarithms is an essential property that helps simplify logarithmic expressions involving division.
When you have an expression of the form \( \log_{a} m - \log_{a} n \), you can use this rule to combine the logarithms into a single log term.
The Quotient Rule formula is as follows:

• \( \log_{a} m - \log_{a} n = \log_{a} \left( \frac{m}{n} \right) \).

This means you take the logarithm of the quotient, where the numerator is represented by \( m \) and the denominator by \( n \).
In our exercise, this rule helps us simplify the expression
after we have already applied the Product and Power Rules.
The existing expression \( \log_{a} (b \cdot d^c) - r \log_{a} s \) can be rewritten as \( \log_{a} \left( \frac{b \cdot d^c}{s^r} \right) \), making it more compact and manageable.
Understanding this rule facilitates smooth manipulation of complex logarithmic expressions.

Product Rule of Logarithms

The Product Rule of Logarithms assists in combining two or more logarithmic terms that are connected by addition.
The rule is most helpful when facing expressions like \( \log_{a} m + \log_{a} n \).
According to the Product Rule, this expression can be simplified as:

• \( \log_{a} m + \log_{a} n = \log_{a} (m \cdot n) \).

This converts the addition of logarithms to the logarithm of a product.
In our problem, the expression \( \log_{a} b + c \log_{a} d \) uses the Product Rule.
Here, \( \log_{a} b \) and \( \log_{a} (d^c) \) add up, forming the compound term \( \log_{a} (b \cdot d^c) \).
Understanding when and how to apply this rule eases the combination of separate logarithmic terms into a single expression.

Power Rule of Logarithms

The Power Rule of Logarithms is a handy tool when dealing with logarithms that include exponents.
It allows you to bring exponents down in front of the logarithm, simplifying the expression.
Here's how the Power Rule works:

• \( c \log_{a} d = \log_{a} (d^c) \).

This rule enables the transformation of a multiplied factor inside a logarithm into an exponent.
In the given exercise, it simplifies parts of the expression such as \( c \log_{a} d \) and converts it into \( \log_{a} (d^c) \).
When combined with other properties like the Product or Quotient Rule, it helps in expressing complicated logarithmic terms more concisely.
Mastering this rule is key to resolving expressions with powers, making them more accessible for further calculations or combination.