Problem 65 - A Graphical Approach to College Algebra

Question

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{3} \frac{2}{5}$$

Step-by-Step Solution

Answer

\( \log_{3} 2 - \log_{3} 5 \)

Step 1: Identify the Property of Logarithms

Recall that the logarithm of a quotient can be expressed as the difference of the logarithms. For any positive real numbers \( a \) and \( b \), and any positive real number \( c \), \( \log_{c} \left( \frac{a}{b} \right) = \log_{c} a - \log_{c} b \).

Step 2: Apply the Quotient Property

Apply the quotient property to the given logarithm: \[\log_{3} \left( \frac{2}{5} \right) = \log_{3} 2 - \log_{3} 5\]

Step 3: Simplified Expression

The expression is now rewritten using the property: \( \log_{3} \left( \frac{2}{5} \right) = \log_{3} 2 - \log_{3} 5 \).This is the simplest form using the properties of logarithms.

Key Concepts

quotient property of logarithms

The quotient property of logarithms is a useful tool for breaking down complex logarithmic expressions. When you have a logarithm of a fraction, you can simplify it by separating it into two individual logarithms that are easier to handle. For instance, if you have the expression \( \log_{c} \left( \frac{a}{b} \right) \), you can rewrite it as \( \log_{c} a - \log_{c} b \). This means that the logarithm of a quotient is equivalent to the difference between the logarithms of the numerator and the denominator.
This property helps when you want to simplify logarithmic expressions that appear complicated at first glance. It's a direct and elegant way to break down fractions inside logarithms into their straightforward components. Next time you see a quotient inside a logarithm, remember you can simplify by subtraction!

logarithmic expressions

Logarithmic expressions involve logarithms, which are mathematical operations that help you determine the power to which a base number must be raised to produce a given number. Understanding how to manipulate these expressions is essential in various mathematical contexts. Typically, a logarithmic expression might look something like \( \log_{b}(x) \), where \( b \) is the base, and \( x \) is the number you want to find the power of. Working with logarithmic expressions often requires identifying patterns and applying properties like the quotient property or the product and power properties.
Logarithms can simplify expressions related to exponential growth, such as populations or investments. Learning to handle logarithmic expressions provides clarity in solving problems where numbers grow or shrink exponentially.

simplifying logarithms

Simplifying logarithms involves using properties like the quotient, product, and power properties to transform complex logarithmic expressions into simpler forms. This makes solving equations or evaluating expressions more manageable. For example, if you're given an expression like \( \log_{3} \left( \frac{2}{5} \right) \), you can use the quotient property to simplify it to \( \log_{3} 2 - \log_{3} 5 \). By writing complex logarithmic expressions in simpler terms, calculations involving logarithms become less cumbersome and more intuitive.
To effectively simplify logarithms, it's important to consistently apply these properties whenever possible. Doing so not only helps in algebraic manipulation but also enhances your understanding of how logarithms relate to their underlying bases.