Problem 13 - College Algebra

Question

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

Step-by-Step Solution

Answer

The expression simplifies to \(\log_{b}(7)\).

Step 1: Apply Logarithm Power Rule

To simplify \(-\log _{b} \left( \frac{1}{7} \right)\), we apply the power rule of logarithms: \(-\log_{b} \left( x \right) = \log_{b} \left( x^{-1} \right)\). Therefore, \(-\log_{b} \left( \frac{1}{7} \right)\) becomes \(\log_{b} \left( \left( \frac{1}{7} \right)^{-1} \right)\).

Step 2: Simplify the Expression Inside the Logarithm

Now, simplify the expression \(\left( \frac{1}{7} \right)^{-1}\). The reciprocal of \(\frac{1}{7}\) is 7, so \(\left( \frac{1}{7} \right)^{-1} = 7\).

Step 3: Write the Final Condensed Form

Substitute back into the logarithm to get the final simplified form: \(\log_{b}(7)\). Thus, the expression\(-\log\left(\frac{1}{7}\right)\) is condensed into a single logarithm \(\log_{b}(7)\).

Key Concepts

Logarithm Power Rule

The logarithm power rule is a handy simplification technique in mathematics that helps streamline complex logarithmic expressions. Logarithms, in essence, are the "inverses" of exponential functions, and the power rule takes advantage of this relationship. The power rule states that:

• \(-\log_{b}(x) = \log_{b}(x^{-1})\)

If you encounter a negative logarithm, you can change it into a positive one by taking the reciprocal of the argument inside the logarithm. This approach is particularly useful for transforming and simplifying expressions involving negative logarithms.
For example, in the expression \(-\log_{b}\left(\frac{1}{7}\right)\), applying the power rule changes it to \(\log_{b}\left(\left(\frac{1}{7}\right)^{-1}\right)\). Through this, negative signs are swapped for reciprocals, making the expression easier to manage.
Understanding and applying the logarithm power rule can greatly simplify mathematical expressions, particularly in algebra and calculus.

Condensing Logarithms

Condensing logarithms involves combining multiple logarithmic expressions into a single logarithm, using logarithmic laws to make expressions simpler to handle. Condensing is useful for both simplifying expressions and solving logarithmic equations. The main rules used for condensing include:

• The product rule: \(\log_{b}(MN) = \log_{b}(M) + \log_{b}(N)\)
• The quotient rule: \(\log_{b}\left(\frac{M}{N}\right) = \log_{b}(M) - \log_{b}(N)\)
• The power rule: \(\log_{b}(M^n) = n\cdot\log_{b}(M)\)

In our exercise, we start with \(-\log_{b}\left(\frac{1}{7}\right)\). After applying the logarithm power rule, we can condense it to a single logarithm, \(\log_{b}(7)\), showing that not all simplifications require the straightforward combination of terms but rather clever application of rules.
By combining and reworking these rules, you can often take expressions with multiple logarithms and reduce them to a single term, simplifying the problem and making it more approachable.

Reciprocal in Mathematics

In mathematics, the reciprocal of a number is essentially "1 divided by" that number. It's a crucial concept in arithmetic and algebra, simplifying equations and expressions significantly. Reciprocal is denoted as follows:

• If your number is \(x\), then its reciprocal is \(\frac{1}{x}\).

The reciprocal rule comes into play when dealing with powers, as we observed in the presented solution: \(\left(\frac{1}{7}\right)^{-1} = 7\). A reciprocal "flips" a fraction, effectively transforming the numerator into the denominator and vice versa.
When using reciprocals in logarithmic expressions, like transforming \(-\log_{b}\left(\frac{1}{7}\right)\) into \(\log_{b}(7)\), it bridges gaps between seemingly complex mathematical steps.
By comprehending this simple yet powerful concept, you're well-equipped to tackle a variety of mathematical problems, particularly those involving negative exponents or logarithmic expressions.