Problem 74 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of logarithms to condense the expression.$$\frac{5}{2} \log _{7}(z-4)$$.

Step-by-Step Solution

Answer

The condensed form of the given logarithmic expression is \(\log_7{(z - 4)^{5/2}}\)

Step 1: Apply the property of logarithms for coefficients

By applying the power rule of logarithms, the coefficient \(5/2\) at the front can be applied as an exponent to the argument to get the following expression: \(\log_7{(z - 4)^{5/2}}\)

Step 2: There's no more simplification necessary

Checking the result, the task of condensing the logarithmic expression is already accomplished. No further steps or simplifications are needed.

Key Concepts

Logarithmic Expressions

Logarithmic expressions involve expressions that contain logarithms. They serve as the reverse operations of exponentiation. If you have a logarithmic expression like \( \log_b(a) \), this reads as "the power to which the base \( b \) must be raised to produce \( a \)." Understanding this fundamental can simplify working with them. A common operation is to change the structure of these expressions using properties of logarithms, like the product, quotient, and power rules. Each of these brings a different flexibility when dealing with logs, enabling various transformations to simplify or condense expressions.

Condensing Logarithms

Condensing logarithms is the process of combining multiple logarithmic terms into a single logarithm. This is particularly useful when simplifying expressions or solving logarithmic equations. The properties of logarithms come in handy here. One important property used in condensing is the Power Rule. For instance, if you have an expression like \( n \log_b(a) \), it can be condensed to \( \log_b(a^n) \) as per the power rule. Condensing simplifies the expression and can help in identifying solutions more easily. Moreover, it can make complex algebra involving logarithms easier to handle.

Power Rule of Logarithms

The Power Rule of logarithms is a vital property that states \( n \log_b(a) = \log_b(a^n) \). It allows us to move a coefficient in front of a logarithm as an exponent of the argument inside the logarithmic function. This rule is useful for turning a cumbersome logarithmic expression into a simpler form. For example, in the exercise provided, \( \frac{5}{2} \log_7(z-4) \) can be rewritten using the power rule as \( \log_7((z-4)^{5/2}) \). This transformation is essential for tasks like condensing because it helps reduce the expression to one single logarithmic term.