Problem 25 - Algebra 2

Question

Expand each logarithm. \(4 \log m-\log n\)

Step-by-Step Solution

Answer

The expanded form of the given logarithm \(4 \log m - \log n\) is \(\log\frac{m^4}{n}\).

Step 1: Apply the Power Rule

The first term has a coefficient of 4 attached to the logarithm. Applying the power rule to that logarithm gives \((\log m^4 - \log n)\).

Step 2: Simplify using properties of logarithms

Now we can write this as a single logarithm using the quotient property of logarithms, which is \((\log\frac{m^4}{n})\). Note that this property can only be used when the logarithmic terms are subtracted.

Key Concepts

Power Rule in Logarithms

When working with logarithmic expressions, understanding the Power Rule can be very beneficial. The Power Rule states that when you have a logarithm with an exponent, you can move the exponent to the front as a coefficient. This simplifies the expression significantly.

Here's what the Power Rule looks like in mathematics:

• If you have \(\log_b(a^c)\), it can be rewritten as \(c \log_b(a)\).

In the given exercise, we applied the Power Rule to the term \(4 \log m\). This means we could rewrite it as \(\log(m^4)\). By doing this, we eliminated the coefficient and simplified our work.

Always remember that using the Power Rule allows us to easily simplify logarithmic expressions where multipliers are involved. This can particularly be useful when dealing with more complicated expressions or when combining logarithms.

Quotient Property of Logarithms

The Quotient Property of Logarithms is another essential concept when simplifying logarithmic expressions. This property lets us express the subtraction of two logarithms as a single logarithm by using division.

Here's how it works:

• When you have \(\log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right)\).

This property relies on the arithmetic operation of division to combine the logarithms.

In our example, once we used the Power Rule, we ended up with \(\log(m^4) - \log(n)\). By applying the Quotient Property, we could express this as a single logarithm: \(\log\left(\frac{m^4}{n}\right)\).

Understanding and applying this property is crucial as it allows us to condense expressions, making them easier to interpret and solve. The property is especially useful in equations and real-life applications where processes naturally involve percentages, ratios, or comparisons.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions often involves combining various properties and rules, such as the Power Rule and the Quotient Property, to rewrite the expression in the simplest form possible. Simplification makes solving equations more straightforward and helps to better understand the relationships within the data.

Let's break down the simplification process:

• Identify parts of the expression that can benefit from Power Rule or Quotient Property.
• Rewrite each applicable part using these rules, as shown in the Power Rule and Quotient Property sections.
• Combine the logarithms into a single expression where possible, using further simplifications like combining fractions or factoring.

In the given exercise, we started with \(4 \log m - \log n\). First, applying the Power Rule led to \(\log(m^4)\). Then, using the Quotient Property allowed us to combine it all into \(\log\left(\frac{m^4}{n}\right)\), simplified into one neat expression.

Simplifying expressions is an essential skill that not only helps in exams and textbooks but also is extremely valuable in real-world scenarios where complex mathematical models are involved.