Problem 65 - Algebra and Trigonometry

Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \frac{1}{2}\left(\log _{5} x+\log _{5} y\right)-2 \log _{5}(x+1) $$

Step-by-Step Solution

Answer

The given expression simplifies to \(log_{5}\sqrt{\frac{xy}{(x+1)^2}}\)

Step 1: Apply properties of logarithms

Apply the product rule to \(\frac{1}{2}(\log _{5} x+\log _{5} y)\) to condense it. This gives \(\frac{1}{2}\log _{5}(xy)\). Then apply the power rule to \(2 \log _{5}(x+1)\) moving the coefficient to be the exponent, thus \(log_{5}((x+1)^2)\). The new expression becomes \(\frac{1}{2}\log _{5}(xy) - \log _{5}((x+1)^2)\).

Step 2: Use properties of logarithms to condense further

Apply the quotient rule which states that the difference between two logs with the same base equals the log of the quotient of their arguments. So, \(\frac{1}{2}\log _{5}(xy) - \log _{5}((x+1)^2)\) becomes \(log_{5}(\frac{xy}{(x+1)^2})\).

Step 3: Simplify logarithmic expression

Use the power rule again to remove the factor of \(\frac{1}{2}\). This transforms the expression to \(log_{5}\sqrt{\frac{xy}{(x+1)^2}}\).

Step 4: Simplify further for final form

Since we can't simplify the log any further without specific values for x and y, our final form is \(log_{5}\sqrt{\frac{xy}{(x+1)^2}}\)

Key Concepts

Logarithm Condensation

Logarithm condensation is a powerful technique in the simplification of logarithmic expressions. It involves combining multiple logarithms into a single logarithm. This not only makes the expression neater, but it also often paves the way for further simplifications and solutions. For instance, if we have separate logarithms being added or subtracted, they could represent a multiplication or division within a single logarithm after condensation.

When solving the exercise, the properties of logarithms help in managing the coefficients and condensing the terms into one. Initially, the expression consists of two separate logarithmic functions. First, we combine these using the product rule, which leads to a multiplication inside the logarithm. Then, after applying the power rule, we condense the expression further using the difference of the logarithms, leading to a single log that represents a quotient. This single log is much simpler and can be more easily processed for further mathematical operations.

Product Rule of Logarithms

The product rule of logarithms states that the log of a product is equal to the sum of the logs of the factors. Mathematically expressed as: \[\log_b(mn) = \log_b(m) + \log_b(n)\].

This rule is fundamental in logarithm condensation, as it allows us to combine separate logarithmic terms that are being added into one term. In our exercise, this property was used in the first step to combine \(\frac{1}{2}(\log _{5} x + \log _{5} y)\) resulting in \(\frac{1}{2}\log _{5}(xy)\). The coefficient was managed to ensure the expression remained a single logarithm, maintaining the condensation process.

Quotient Rule of Logarithms

On the flip side, the quotient rule of logarithms addresses how to deal with division within a logarithmic function. It indicates that the log of a quotient is the difference of the logs of the numerator and denominator. The formula is: \[\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\].

In the given exercise, this rule was used to condense the expression further in Step 2. After applying the product and power rules, the difference of \(\frac{1}{2}\log _{5}(xy)\) and \(\log _{5}((x+1)^2)\) was expressed as a single logarithm, \(\log_{5}(\frac{xy}{(x+1)^2})\), with the entire argument under the same log function.

Power Rule of Logarithms

The power rule of logarithms is essential when dealing with expressions that include logarithms of numbers raised to a power. It states that the log of a number raised to an exponent is equal to the exponent times the logarithm of the number itself: \[\log_b(m^n) = n\cdot \log_b(m)\].

In the textbook problem, this property was cleverly used twice. Initially, it was used to adjust the logarithmic expression, moving the coefficient of 2 in front of the log to the exponent of its argument. Later on, in the final step (Step 3), it was once again utilized to adjust the \(\frac{1}{2}\) factor, ultimately leading to the expression \(\log_{5}\sqrt{\frac{xy}{(x+1)^2}}\). The power rule is instrumental in simplifying logarithms and getting a polished, easier-to-read logarithmic form.