Problem 68 - Algebra 2

Question

Write each expression as a single logarithm. \(\log _{3} y+4 \log _{3} t\)

Step-by-Step Solution

Answer

The single logarithm representation of the given expression is \( \log_3{y*t^4} \).

Step 1: Apply the power rule on the second logarithmic term

The power property of logarithm states that \(a \log_b m = \log_b m^a\). So, \(4 \log_{3} t\) will become \(\log_3{t^4}\)

Step 2: Apply the product rule on the entire expression

The product property of logarithm states that \(\log_b m + \log_b n = \log_b mn\). So, \(\log_3 y + \log_3{t^4}\) will become \(\log_3(y*t^4)\)

Step 3: Simplify the expression

Simplify the expression to get the final answer. In this case, no further simplification is necessary, and the solution \(\log_3(y*t^4)\) is already in its simplest form

Key Concepts

Power Rule

The power rule is a method used in logarithms to simplify expressions by converting coefficients into exponents. When you have a logarithmic expression like \( a \log_b m \), you can use the power rule to transform it into \( \log_b m^a \). For example, in the expression \( 4 \log_3 t \), the coefficient \( 4 \) in front of the log indicates that you're multiplying \( \log_3 t \) by 4. By applying the power rule, you shift this multiplication to make it an exponent, converting the expression into \( \log_3 t^4 \). This is very helpful when you want to combine logarithmic terms because it simplifies the number of terms you're working with.

• The power rule allows us to handle coefficients effectively.
• By simplifying coefficients into exponents, it makes combining terms easier.

When working with logarithmic expressions, being comfortable with the power rule is key.

Product Rule

The product rule in logarithms is used to combine two or more log terms into a single term. If you have two logs with the same base, like \( \log_b m \) and \( \log_b n \), the product rule allows you to combine them into one by multiplying their arguments: \( \log_b mn \).Let's apply this to the expression \( \log_3 y + \log_3 t^4 \). Here, both logs have the same base, 3, allowing us to seamlessly combine them using the product rule. This results in a single log: \( \log_3(y \cdot t^4) \).

• The product rule only works with terms that have the same base.
• It helps in simplifying and reducing the number of logarithmic expressions.
• Combined expressions are more compact and easier to interpret.

Using the product rule makes managing logarithmic expressions more straightforward, especially when simplifying or solving equations.

Simplifying Expressions

Simplifying expressions is the process of reducing an expression to its simplest form. This often involves combining like terms and applying mathematical properties to make the expression as concise as possible. In the context of logarithmic expressions, simplification can mean reducing a multi-term expression to a single log term.Once you have used rules like the power and product rules, your expression may already be in its simplest form, as seen in our example where \( \log_3(y \cdot t^4) \) was obtained.

• Simplifying helps in making expressions more readable and manageable.
• It is essential in problem-solving as it provides clarity.
• This ensures that no further mathematical manipulation is required.

Always check if further simplification is possible, but in many cases, like the one here, the expression might already be at its simplest.