Problem 54 - Algebra and Trigonometry

Question

\(45-54\) . 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 Power Rule

The given expression is \( \log_{a} b + c \log_{a} d - r \log_{a} s \). First, use the power rule of logarithms, which states \( c \log_{a} d = \log_{a} d^c \) and \( r \log_{a} s = \log_{a} s^r \). Transform the expression as follows:\[\log_{a} b + \log_{a} d^c - \log_{a} s^r\]

Step 2: Apply the Product Rule

Now, use the product rule of logarithms, which states \( \log_{a} x + \log_{a} y = \log_{a} (xy) \). Apply this to combine the first two terms:\[\log_{a} (b \cdot d^c)\]

Step 3: Apply the Quotient Rule

Finally, use the quotient rule of logarithms, which states \( \log_{a} x - \log_{a} y = \log_{a} \left(\frac{x}{y}\right) \). Subtract the last term:\[\log_{a} \left(\frac{b \cdot d^c}{s^r}\right)\]This is the simplified form of the expression.

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a very useful tool when solving logarithmic expressions, especially when you need to simplify expressions or solve logarithmic equations.This rule says that if you have a logarithm where a number is multiplied by a logarithmic term, such as \( c \log_{a} d \), you can transform it into an exponent form: \( \log_{a} d^c \). What this basically means is that you can "bring up" the multiplier in front of the log and make it an exponent of the argument inside the logarithm. For example:

• \( 3 \log_{7} x = \log_{7} x^3 \)
• \( 2 \log_{10} y = \log_{10} y^2 \)

This transformation makes it easier to apply other logarithm rules later on, like the product or quotient rule. Practice applying the power rule by moving coefficients to exponents, and soon you’ll find it becomes second nature.

Product Rule of Logarithms

The product rule of logarithms is essential for combining multiple logarithmic expressions into a single term.According to this rule, when you have two terms that are added together, such as \( \log_{a} x + \log_{a} y \), you can combine them by multiplying the inside terms and writing them as one logarithm: \( \log_{a} (xy) \). Essentially, adding logs means multiplying their insides. Here are some examples:

• \( \log_{5} 3 + \log_{5} 4 = \log_{5} (3 \cdot 4) = \log_{5} 12 \)
• \( \log_{2} 7 + \log_{2} 8 = \log_{2} (7 \cdot 8) = \log_{2} 56 \)

Learning to use the product rule helps simplify expressions and is extremely useful in solving equations where multiple terms need to be combined.

Quotient Rule of Logarithms

The quotient rule of logarithms is another crucial tool for simplifying and solving logarithmic expressions.It states that when you subtract one logarithm from another, such as \( \log_{a} x - \log_{a} y \), you can simplify it by turning it into a single logarithm, dividing the inside terms: \( \log_{a} \left(\frac{x}{y}\right) \). So, subtraction in the log world translates into division for the arguments. Take a look at these examples:

• \( \log_{6} 18 - \log_{6} 2 = \log_{6} \left(\frac{18}{2}\right) = \log_{6} 9 \)
• \( \log_{3} 27 - \log_{3} 9 = \log_{3} \left(\frac{27}{9}\right) = \log_{3} 3 \)

Mastering the use of the quotient rule allows you to tackle more complicated logarithmic expressions by simplifying them step by step.