Problem 59
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. $$\frac{1}{2}\left(\log _{5} x+\log _{5} y\right)-2 \log _{5}(x+1)$$
Step-by-Step Solution
Verified Answer
The simplified form of the given expression is \(\log_{5} \left(\frac{\sqrt{xy}}{(x + 1)^2}\right)\)
1Step 1: Apply Logarithmic Rules
First, apply the power rule to the term \(-2 \log _{5}(x+1)\) which converts into \(-\log _{5}(x + 1)^2\). Simultaneously, the product rule of logarithms is applied to the term \(\frac{1}{2}\left(\log _{5} x+\log _{5} y\right)\), which results in the expression \(\log _{5}\sqrt{xy}\).
2Step 2: Simplify the Expression
Next, apply the subtraction rule, which states that the logarithm of a quotient is the difference of the logarithms, to combine the two logs into one, yielding the result \(\log_{5} \left(\frac{\sqrt{xy}}{(x + 1)^2}\right)\)
3Step 3: Evaluate Logarithmic Expression
If \(x\) and \(y\) are known numbers and can be substituted into the expression, carry out the calculations. If they are not available, this is the final expression.
Key Concepts
Logarithmic PropertiesLogarithm Power RuleLogarithm Product RuleLogarithm Quotient Rule
Logarithmic Properties
Logarithms have unique properties that make them extremely useful for simplifying complex mathematical expressions. Understanding these properties is crucial when dealing with logarithmic equations or when trying to condense multiple logarithms into a single term.
Mastering these properties not only helps in solving logarithmic equations but also, as with our exercise, in condensing logarithmic expressions to their simplest forms.
Product Property
When two logs with the same base are added, this translates into one log of the product of the operands: \[\log_b(m) + \log_b(n) = \log_b(m \times n)\].Quotient Property
Conversely, when a logarithm is subtracted from another, the difference represents the quotient of the operands: \[\log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right)\].Power Property
The coefficient of the logarithm can be moved up to become the exponent of the log's argument: \[a \log_b(m) = \log_b(m^a)\].Mastering these properties not only helps in solving logarithmic equations but also, as with our exercise, in condensing logarithmic expressions to their simplest forms.
Logarithm Power Rule
When you encounter logarithms raised to a power or multiplied by a coefficient, the power rule offers the simplification needed. The rule states that the exponent on a logarithmic argument can be brought out front as a multiplier:
\[\log_b(m^n) = n \log_b(m)\].
This can also be applied in reverse to move a coefficient into the exponent position of the logarithm's argument, essential for condensing operations: \[a \log_b(m) = \log_b(m^a)\].
For instance, in the exercise, we have \( -2 \log_5(x+1) \) which according to the power rule simplifies to \( -\log_5((x+1)^2) \). Such transformations are central to understanding and solving more complex logarithmic expressions.
\[\log_b(m^n) = n \log_b(m)\].
This can also be applied in reverse to move a coefficient into the exponent position of the logarithm's argument, essential for condensing operations: \[a \log_b(m) = \log_b(m^a)\].
For instance, in the exercise, we have \( -2 \log_5(x+1) \) which according to the power rule simplifies to \( -\log_5((x+1)^2) \). Such transformations are central to understanding and solving more complex logarithmic expressions.
Logarithm Product Rule
The product rule is particularly useful when facing several logarithms to be combined into one. This property is used when you have two logarithms with the same base being added together:
\[\log_b(m) + \log_b(n) = \log_b(m \times n)\].
In our exercise, this property enables us to condense \(\frac{1}{2}(\log_5 x + \log_5 y)\) into \(\log_5 \sqrt{xy}\), essentially rewriting the addition of logarithms as the logarithm of a product. This step is pivotal because it converges separate log terms into a single coherent term, making it easier to evaluate if needed.
\[\log_b(m) + \log_b(n) = \log_b(m \times n)\].
In our exercise, this property enables us to condense \(\frac{1}{2}(\log_5 x + \log_5 y)\) into \(\log_5 \sqrt{xy}\), essentially rewriting the addition of logarithms as the logarithm of a product. This step is pivotal because it converges separate log terms into a single coherent term, making it easier to evaluate if needed.
Logarithm Quotient Rule
The quotient rule for logarithms offers a pathway to condensing expressions that involve the division of terms. The rule is defined as the difference between two logarithms being the logarithm of their quotient:
\[\log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right)\].
By applying this rule in the context of our exercise, we transform the expression by moving from two separate logs, \(\log_5 \sqrt{xy}\) and \(\log_5 (x+1)^2\), to a single log expression that captures the quotient: \(\log_5\left(\frac{\sqrt{xy}}{(x + 1)^2}\right)\). This step is the essence of 'condensing' where separate logarithmic expressions are melded into one clean logarithmic statement, streamlining the path to a solution.
\[\log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right)\].
By applying this rule in the context of our exercise, we transform the expression by moving from two separate logs, \(\log_5 \sqrt{xy}\) and \(\log_5 (x+1)^2\), to a single log expression that captures the quotient: \(\log_5\left(\frac{\sqrt{xy}}{(x + 1)^2}\right)\). This step is the essence of 'condensing' where separate logarithmic expressions are melded into one clean logarithmic statement, streamlining the path to a solution.
Other exercises in this chapter
Problem 59
Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer.
View solution Problem 59
Evaluate each expression without using a calculator. $$\ln 1$$
View solution Problem 59
In college, we study large volumes of information \(-\) information that, unfortunately, we do not often retain for very long. The function $$f(x)=80 e^{-0.5 x}
View solution Problem 60
Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer.
View solution