Problem 261 - Precalculus

Question

For the following exercises, condense to a single logarithm if possible. $$ -\log _{b}\left(\frac{1}{7}\right) $$

Step-by-Step Solution

Answer

\( -\log_{b}\left(\frac{1}{7}\right) = \log_{b}(7) \)

Step 1: Understand the Expression

The expression given is \[ -\log_{b}\left(\frac{1}{7}\right) \]This represents the negative of a logarithm with base \( b \) of the fraction \( \frac{1}{7} \). The goal is to simplify or condense this expression into a single logarithm form.

Step 2: Use the Property of Negative Logarithm

Recall the property of logarithms that states:\[ -\log_{b}(x) = \log_{b}\left(\frac{1}{x}\right) \]Apply this property to the expression:\[ -\log_{b}\left(\frac{1}{7}\right) = \log_{b}(7) \] The negative sign inverts the argument inside the logarithm.

Step 3: Result

The expression \[ -\log_{b}\left(\frac{1}{7}\right) \]has been condensed to the logarithm \[ \log_{b}(7) \].This is a single logarithm expression without any negative sign.

Key Concepts

Condensing Logarithms

Logarithms can often appear lengthy or complex, but one of the key tasks in algebra is to simplify them, or "condense" them, into a more manageable form. "Condensing logarithms" means combining or transforming log expressions so that they appear as a single logarithm rather than multiple ones. This becomes extremely useful in solving equations or simplifying expressions.
To condense a logarithm, you typically need to employ the three main properties of logarithms:

• Product Property: \(\log_b(x) + \log_b(y) = \log_b(x \cdot y)\)
• Quotient Property: \(\log_b(x) - \log_b(y) = \log_b(\frac{x}{y})\)
• Power Property: \(c\log_b(x) = \log_b(x^c)\)

Using these properties, any expression involving multiple logarithms—with the same base—can be rewritten or condensed into a single expression. Such simplifications often enable easier computation and clearer understanding of mathematical relationships in an expression.

Negative Logarithms

A negative logarithm might seem intimidating, but it's just another opportunity to simplify! When you see a negative sign in front of a log expression, you're essentially dealing with an inverse problem. The property that guides this transformation is:\[ -\log_b(x) = \log_b\left(\frac{1}{x}\right) \]Using this, the negative sign suggests that we take the reciprocal of the argument inside the log without changing the base.
This conversion is particularly helpful when you're trying to transform a negative log into a positive one, thus making it easier to manage in further calculations. Remember, this property is powerful not only in simplifying expressions but also in ensuring that all log values you work with are non-negative, thereby aligning with how logs inherently operate: representing exponent values that are typically non-negative.

Single Logarithm Expression

When it comes to simplifying logs, the ultimate goal often is to express them as a "single logarithm expression." Doing this not only makes equations cleaner but also often makes the next steps in solving an equation much easier.
With all the properties of logarithms and tools like converting negative logs into their positive counterparts using the negative log property, finding a single logarithm expression involves consolidating any operations within the log expression into one cohesive form.For example, the exercise you've encountered walks you through condensing such an expression:\[-\log_{b}\left(\frac{1}{7}\right)\] is initially complex but by applying the negative log property, you simplify it to:\[\log_{b}(7)\] This transformation has turned a seemingly cumbersome negative logarithm into a straightforward positive expression—a single, composite log term that elegantly encapsulates all `\(\frac{1}{7}\)` values at its heart.