Problem 79
Question
Write each logarithmic expression as a single logarithm. $$ \log _{3} 12-\log _{3} 2 $$
Step-by-Step Solution
Verified Answer
The resulting single logarithm is \( \log _{3} 6 \).
1Step 1: Identify the Given Expression
The given expression is \( \log _{3} 12-\log _{3} 2 \). The exercise requirement is to convert the expression into a single logarithm.
2Step 2: Apply Logarithm Properties
Let's apply the rule of logarithms that states \(\log _{b} x - \log _{b} y = \log _{b}(x / y)\). This rule allows dividing 12 by 2, because there are two logarithms with the same base being subtracted.
3Step 3: Simplify The Expression
Calculate the value inside the logarithm that is 12 divided by 2 which equals to 6. Write down the result as \( \log _{3} 6 \).
Key Concepts
Logarithmic ExpressionsLogarithm PropertiesSimplifying Logarithms
Logarithmic Expressions
Logarithmic expressions represent numbers through their relationship to a given base raised to a power. In simpler terms, a logarithm tells us what power we need to raise a base number to get a specific result. For example, in the expression \( \log_{3} 12 \), the base is 3. We are trying to understand what power, when applied to 3, will give us 12.
To express numbers as logarithms can be very useful in scenarios involving exponential growth, compound interest, or scientific computations. They are a way of expressing large numbers in compact form, which simplifies many calculations. Instead of working with large numbers directly, we work with their exponents.
To express numbers as logarithms can be very useful in scenarios involving exponential growth, compound interest, or scientific computations. They are a way of expressing large numbers in compact form, which simplifies many calculations. Instead of working with large numbers directly, we work with their exponents.
Logarithm Properties
Logarithms have some handy properties that allow us to work with them more easily. One of the most important properties is the subtraction property, which was used in the solution to the original exercise. This property is given by:
Other important properties include the addition property \(\log_{b} x + \log_{b} y = \log_{b} (xy)\) and the power property \(\log_{b} (x^n) = n \log_{b} x\). Each of these properties comes into play when simplifying complex expressions and solving logarithmic equations. Understanding these properties is key to being comfortable working with any logarithmic expression.
- \(\log_{b} x - \log_{b} y = \log_{b} \left( \frac{x}{y} \right)\)
Other important properties include the addition property \(\log_{b} x + \log_{b} y = \log_{b} (xy)\) and the power property \(\log_{b} (x^n) = n \log_{b} x\). Each of these properties comes into play when simplifying complex expressions and solving logarithmic equations. Understanding these properties is key to being comfortable working with any logarithmic expression.
Simplifying Logarithms
Simplifying logarithms often involves using the properties of logarithms to make an expression easier to work with or understand. In the step-by-step solution provided, the subtraction property was used to simplify \( \log_{3} 12 - \log_{3} 2 \) into a single logarithm: \( \log_{3} 6 \).
The simplification process requires identifying opportunities to apply these logarithm properties. For subtraction, as seen here, after recognizing the shared base and applying it, you'll likely need to calculate the division of the two numbers involved. The goal of simplification is to reduce the expression to its most compact and manageable form, making it easier to interpret or further manipulate. Remember, simplifying correctly allows us to unravel complex equations and solve real-world problems efficiently.
Whenever faced with logarithms, a quick check for which properties you can apply can save a lot of time and ensure your solutions remain accurate.
The simplification process requires identifying opportunities to apply these logarithm properties. For subtraction, as seen here, after recognizing the shared base and applying it, you'll likely need to calculate the division of the two numbers involved. The goal of simplification is to reduce the expression to its most compact and manageable form, making it easier to interpret or further manipulate. Remember, simplifying correctly allows us to unravel complex equations and solve real-world problems efficiently.
Whenever faced with logarithms, a quick check for which properties you can apply can save a lot of time and ensure your solutions remain accurate.
Other exercises in this chapter
Problem 78
Write each logarithmic expression as a single logarithm. $$ \log 3+\log 5 $$
View solution Problem 78
For each rational function, find any points of discontinuity. $$ y=\frac{2}{x+1} $$
View solution Problem 79
For each rational function, find any points of discontinuity. $$ y=\frac{1}{x^{2}-5 x+6} $$
View solution Problem 80
Write each logarithmic expression as a single logarithm. $$ 3 \log 2-\log 4 $$
View solution