Problem 32
Question
Write each as a single logarithm. Assume that variables represent positive numbers. $$ \log _{8} 5+\log _{8} 15-\log _{8} 20 $$
Step-by-Step Solution
Verified Answer
\(\log_8 \left(\frac{15}{4}\right)\)
1Step 1: Use the Product Rule for Logarithms
The product rule states that \(\log_b x + \log_b y = \log_b (xy)\). This can be applied to the terms \(\log_8 5\) and \(\log_8 15\). Applying the product rule gives us: \(\log_8 (5 \times 15)\).
2Step 2: Simplify the Product
Simplify the expression inside the logarithm: \(5 \times 15 = 75\). Hence, the expression becomes \(\log_8 75\).
3Step 3: Use the Quotient Rule for Logarithms
The quotient rule states that \(\log_b x - \log_b y = \log_b \left(\frac{x}{y}\right)\). Apply this to combine \(\log_8 75\) and \(\log_8 20\): \(\log_8 \left(\frac{75}{20}\right)\).
4Step 4: Simplify the Quotient
Calculate the simplified form of the quotient inside the logarithm. Simplifying \(\frac{75}{20}\) gives \(\frac{15}{4}\). The expression becomes \(\log_8 \left(\frac{15}{4}\right)\).
5Step 5: Final Expression as a Single Logarithm
We've simplified the original expression to a single logarithm. The final answer is \(\log_8 \left(\frac{15}{4}\right)\).
Key Concepts
Logarithm RulesProduct Rule of LogarithmsQuotient Rule of LogarithmsSimplifying Logarithmic Expressions
Logarithm Rules
Logarithms are a fundamental concept in mathematics used to solve equations involving exponents. Understanding them involves grasping key rules that simplify complex expressions. These logarithm rules allow us to manipulate and combine multiple logarithmic terms into simpler ones, often expressed as a single logarithm.
There are several main rules for logarithms:
There are several main rules for logarithms:
- Product Rule: This is used to combine the logarithms of products into a single logarithm. It states that \( \log_b x + \log_b y = \log_b (xy) \).
- Quotient Rule: This helps in combining logarithms of quotients. It states that \( \log_b x - \log_b y = \log_b \left(\frac{x}{y}\right) \).
Product Rule of Logarithms
The product rule of logarithms is a useful tool when dealing with the sum of logarithms. According to the rule, if you have two logarithms with the same base being added together, you can combine them into a single logarithm. This rule states \( \log_b x + \log_b y = \log_b (xy) \).
In our exercise, we applied the product rule to simplify \( \log_8 5 + \log_8 15 \) into \( \log_8 (5 \times 15) \).
By combining these terms, you directly work with the product rather than separately dealing with each logarithmic expression, making the entire calculation process more straightforward.
In our exercise, we applied the product rule to simplify \( \log_8 5 + \log_8 15 \) into \( \log_8 (5 \times 15) \).
By combining these terms, you directly work with the product rather than separately dealing with each logarithmic expression, making the entire calculation process more straightforward.
Quotient Rule of Logarithms
The quotient rule of logarithms assists in simplifying the subtraction of two logarithms with the same base. This rule states \( \log_b x - \log_b y = \log_b \left(\frac{x}{y}\right) \).
In the given exercise, after using the product rule, we were left with \( \log_8 75 \). The next step was to apply the quotient rule to incorporate \( \log_8 20 \), resulting in \( \log_8 \left(\frac{75}{20}\right) \).
Using the quotient rule effectively helps in transforming the difference between two log terms into a single term, simplifying computations significantly.
In the given exercise, after using the product rule, we were left with \( \log_8 75 \). The next step was to apply the quotient rule to incorporate \( \log_8 20 \), resulting in \( \log_8 \left(\frac{75}{20}\right) \).
Using the quotient rule effectively helps in transforming the difference between two log terms into a single term, simplifying computations significantly.
Simplifying Logarithmic Expressions
Simplifying logarithmic expressions means transforming them into their simplest form, often a single logarithm expression. This involves applying applicable logarithm rules, such as the product and quotient rules, to consolidate the terms.
In our worked example, we simplified the original expression \( \log_8 5 + \log_8 15 - \log_8 20 \) by first applying the product rule to the addends. This gave us \( \log_8 75 \).
Next, the quotient rule was employed to combine \( \log_8 75 \) and \( \log_8 20 \), further simplifying it to \( \log_8 \left(\frac{75}{20}\right) \).
Finally, simplifying the fraction \( \frac{75}{20} \) reduced it to \( \frac{15}{4} \), resulting in the simplest form \( \log_8 \left(\frac{15}{4}\right) \). This reflects the power of logarithm rules in breaking down and simplifying complex expressions.
In our worked example, we simplified the original expression \( \log_8 5 + \log_8 15 - \log_8 20 \) by first applying the product rule to the addends. This gave us \( \log_8 75 \).
Next, the quotient rule was employed to combine \( \log_8 75 \) and \( \log_8 20 \), further simplifying it to \( \log_8 \left(\frac{75}{20}\right) \).
Finally, simplifying the fraction \( \frac{75}{20} \) reduced it to \( \frac{15}{4} \), resulting in the simplest form \( \log_8 \left(\frac{15}{4}\right) \). This reflects the power of logarithm rules in breaking down and simplifying complex expressions.
Other exercises in this chapter
Problem 32
Solve. $$ 125^{x-2}=25 $$
View solution Problem 32
Find \(f(x)\) and \(g(x)\) so that the given function \(h(x)=(f \circ g)(x)\). $$ h(x)=|x-1| $$
View solution Problem 33
Solve. The population of the Cook Islands is decreasing according to the formula \(y=y_{0} e^{-0.0277 t}\). In this formula, \(t\) is time in years and \(y_{0}\
View solution Problem 33
Find the value of each logarithmic expression. $$ \log _{25} 5 $$
View solution