Problem 72 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of logarithms to condense the expression.$$\log _{5} 8-\log _{5} t$$.

Step-by-Step Solution

Answer

The condensed form of the logarithmic expression is \( \log_{5} \frac{8}{t} \).

Step 1: Identify the Logarithmic Expression

We need to evaluate or simplify: \(\log _{5} 8-\log _{5} t\).

Step 2: Apply Logarithm Rules

We use the properties of logarithms: \(\log_b(xy) = \log_b x + \log_b y\), \(\log_b(x/y) = \log_b x - \log_b y\), \(\log_b(x^n) = n\log_b x\).

Step 3: Simplify and Compute

Applying these rules step by step.

Step 4: State the Result

The condensed form of the logarithmic expression is \( \log_{5} \frac{8}{t} \).

Key Concepts

Quotient Rule for Logarithms

Logarithms can often be intimidating, but one of their properties, known as the Quotient Rule, makes them much more approachable. In essence, the Quotient Rule for Logarithms allows us to simplify the logarithm of a division into the difference of two logarithms. Specifically, if you have an expression like \( \log_b{\frac{M}{N}} \), using the Quotient Rule, it can be rewritten as \( \log_b{M} - \log_b{N} \). This property is particularly useful when you want to break down complex logarithmic expressions or combine multiple logarithms into a single term.

Think of the Quotient Rule as a math translator that takes a fraction inside a log and separates it into a subtraction problem outside the log. It’s a nifty trick for condensing or expanding logarithmic expressions for easier calculation or interpretation. When applied to actual problems, like in the given exercise with base 5, it streamlines the process and makes the problem more manageable.

Logarithmic Expressions

A logarithmic expression is essentially a representation of a logarithm, which is an inverse operation to exponentiation. These expressions tell us what power a certain base must be raised to in order to produce a given number. The expression \( \log_b{x} \) is read as 'log base b of x.' This format represents the power to which the base \( b \) must be raised to get \( x \).

Understanding logarithmic expressions is critical because they pop up in various areas of mathematics, from solving exponential equations to modeling growth and decay in real-world scenarios. They have unique properties that allow us to manipulate and re-write them in various ways to simplify complex problems, as demonstrated with the Quotient Rule. Always keep in mind the base of the logarithmic expression since all calculations depend on this underlying value.

Condensing Logarithms

Condensing logarithms is the process of combining multiple logarithmic terms into a single term. This is often done by using logarithmic properties like the Product Rule, Quotient Rule, and Power Rule. In the context of our exercise, we used the Quotient Rule to condense the expression by turning subtraction into division within the logarithm.

Condensing is particularly useful when you encounter multiple logarithmic terms in an equation and you need to simplify the equation for solving. It streamiles the process and can make it easier to visualize what the equation actually represents. Always ensure that the terms you are condensing are using the same base; otherwise, the properties won't apply.

Log Base 5

When we talk about 'log base 5,' we are referring to a logarithmic expression with 5 as the base. This base is important because it dictates how the log will behave. In our exercise, when we see \( \log_5{8} \) and \( \log_5{t} \) being condensed, we’re considering how many times 5 must be multiplied by itself to yield 8 and \( t \) respectively.

Log base 5 appears frequently in certain types of mathematical problems, including exponential growth models and geometric sequences where 5 is a recurring factor. Getting comfortable with various bases, especially common ones like 5, ensures you'll be ready to tackle a wider range of logarithmic problems.