Problem 46 - College Algebra

Question

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \log _{3} 405-\log _{3} 5 $$

Step-by-Step Solution

Answer

The simplified single logarithmic expression is \( log_3(81) \), which evaluates to \(4\).

Step 1: Identify the Operation

Identify the operation taking place between the two logarithms. In this case, it's subtraction. This corresponds to the quotient rule of logarithms.

Step 2: Apply the Rule

Apply the quotient rule, which says that \( log_b(m) - log_b(n) = log_b(m/n) \). Plug in the given logs into this formula to get \( log_3(405) - log_3(5) = log_3(405 / 5) \).

Step 3: Calculate the Quotient

Now, just calculate the quotient inside the parenthesis of the new single logarithm. This gives us \( log_3(81) \).

Step 4: Evaluate the Logarithmic Expression

In this case, it's possible to evaluate the logarithmic expression. Since \(3^4\) is \(81\), the value of \( log_3(81) \) is \(4\).

Key Concepts

Logarithmic Expressions

Understanding how logarithmic expressions work is fundamental in grasping the overall concept of logarithms. A logarithmic expression like \( \text{log}_b(x) \), reads as 'the power to which the base \( b \) must be raised to obtain \( x \)'.

When dealing with expressions that include multiple logarithms, it's important to understand the properties that allow us to simplify or condense these expressions into a single logarithm. This process simplifies complex problems and makes evaluating possible, especially when working without a calculator.

Properties such as the Product Rule, which combines logs with addition into a single log with multiplication, and the Quotient Rule, which does the same for logs with subtraction, are the key to manipulating logarithmic expressions efficiently. Furthermore, recognizing when a log can be evaluated by relating it back to the definition of a logarithm is a skill that comes with practice and a solid understanding of exponents.

Quotient Rule of Logarithms

The quotient rule is particularly useful when dealing with subtraction of two logarithms with the same base. The rule states \( \text{log}_b(m) - \text{log}_b(n) = \text{log}_b\left(\frac{m}{n}\right) \).

Understanding this property allows for the condensing of logarithmic expressions. To apply this rule effectively, always ensure that the base of the logarithms is the same. This rule simplifies the process of subtraction of logs to division inside a single log. It's a powerful tool for expressing ratios of values within logarithmic functions and prepares the expression for potential evaluation if the argument of the resulting log is a power of the base.

Condensing Logarithms

Condensing logarithms is the process of combining multiple logarithmic terms into a single term. This follows from properties like the Quotient Rule for subtraction, as well as the Product Rule for addition, and can also involve the Power Rule when we multiply a logarithm by a coefficient.

Condensing is an essential skill in mathematics and is often required to simplify expressions for easier computation or to set up equations for solving logarithmic equations. To successfully condense logarithms, one must carefully determine which properties apply to the given expression and methodically apply them to condense the expression step by step.

Evaluating Logarithms

Evaluating logarithms means calculating the exact value of a logarithmic expression, if possible. In the context of our example, \( \text{log}_3(81) \), we determined that \( 81 = 3^4 \), so \( \text{log}_3(81) = 4 \) since the base of the logarithm (3) raised to the power of the result gives the number (81).

Sometimes, it's not possible to find an exact value easily, particularly when the result is not an integer. In those cases, approximation techniques or calculators can be used. Being able to evaluate logarithms is essential in many fields, including physics, engineering, and finance, as it is often necessary to solve for unknown exponents.