Problem 26
Question
Write each as a single logarithm. Assume that variables represent positive numbers. $$ 2 \log _{3} 5+\log _{3} 2 $$
Step-by-Step Solution
Verified Answer
\(\log_{3} 50\)
1Step 1: Apply the Power Rule
The Power Rule of logarithms allows us to move a coefficient in front of a log to be an exponent of the argument. Here, apply the Power Rule to the term \(2 \log_{3} 5\), rewriting it as \(\log_{3} (5^2)\). This simplifies to \(\log_{3} 25\). The expression now reads: \(\log_{3} 25 + \log_{3} 2\).
2Step 2: Apply the Product Rule
The Product Rule of logarithms states that the log of a product is equal to the sum of the logs of its factors: \(\log_{b}(xy) = \log_{b}x + \log_{b}y\). Here, use this rule to combine \(\log_{3} 25\) and \(\log_{3} 2\) into a single logarithm: \(\log_{3} (25 \times 2)\).
3Step 3: Simplify the Expression
Multiply the numbers inside the logarithm: \(25 \times 2 = 50\). Therefore, the expression \(\log_{3} (25 \times 2)\) simplifies to \(\log_{3} 50\).
Key Concepts
Power Rule of LogarithmsProduct Rule of LogarithmsSimplifying Expressions
Power Rule of Logarithms
Understanding logarithms begins with learning the fundamental rules, one of which is the Power Rule of Logarithms. The Power Rule is a helpful tool for simplifying expressions where a logarithm is multiplied by a coefficient. This rule allows us to transform an equation of the form \(c \log_b(x)\) into \(\log_b(x^c)\). Essentially, you take the coefficient and make it an exponent of the argument of the logarithm.
In the given exercise, the expression \(2 \log_3 5\) means the coefficient "2" is in front of the logarithm. Using the Power Rule, you convert it to \(\log_3(5^2)\) which simplifies to \(\log_3 25\). This transformation makes it easier to combine this expression with other logs. The main idea here is that multiplying a number by a logarithm is equivalent to raising the logarithm's argument to the power of the number.
In the given exercise, the expression \(2 \log_3 5\) means the coefficient "2" is in front of the logarithm. Using the Power Rule, you convert it to \(\log_3(5^2)\) which simplifies to \(\log_3 25\). This transformation makes it easier to combine this expression with other logs. The main idea here is that multiplying a number by a logarithm is equivalent to raising the logarithm's argument to the power of the number.
Product Rule of Logarithms
The Product Rule of Logarithms is another vital rule when dealing with multiple logarithms. It simplifies the addition of logs into a single logarithm by combining their arguments. The formula is \(\log_b(x) + \log_b(y) = \log_b(xy)\).
When you apply this to the problem, you have two separate logs: \(\log_3 25\) and \(\log_3 2\). The Product Rule lets you combine these into one: \(\log_3(25 \times 2)\). This rule effectively translates the operation of addition from the logarithmic domain back to multiplication in the normal numerical domain. This is useful because it means you can express several different logs as a single log, simplifying calculations and making further analysis easier.
When you apply this to the problem, you have two separate logs: \(\log_3 25\) and \(\log_3 2\). The Product Rule lets you combine these into one: \(\log_3(25 \times 2)\). This rule effectively translates the operation of addition from the logarithmic domain back to multiplication in the normal numerical domain. This is useful because it means you can express several different logs as a single log, simplifying calculations and making further analysis easier.
Simplifying Expressions
Simplifying expressions is all about making math easier and more elegant. After using the Power and Product Rules of Logarithms, you might end up with a single logarithm that can be simplified further. In the case you have a product operation inside the logarithm, this can be calculated straightforwardly.
For example, after applying the rules in the exercise, you get \(\log_3(25 \times 2)\). It simplifies to \(\log_3 50\) as you multiply 25 and 2 to reach 50. The goal of simplifying is to ensure the expression is as straightforward as possible, eliminating any complicated terms. Doing so not only makes equations neater but also easier to interpret and solve. Overall, simplifying is about breaking down or combining elements into a clearer form.
For example, after applying the rules in the exercise, you get \(\log_3(25 \times 2)\). It simplifies to \(\log_3 50\) as you multiply 25 and 2 to reach 50. The goal of simplifying is to ensure the expression is as straightforward as possible, eliminating any complicated terms. Doing so not only makes equations neater but also easier to interpret and solve. Overall, simplifying is about breaking down or combining elements into a clearer form.
Other exercises in this chapter
Problem 26
Each of the following functions is one-to-one. Find the inverse of each function and graph the function and its inverse on the same set of axes. $$ f(x)=4 x+9 $
View solution Problem 26
Solve. $$ 9^{2 x+1}=81 $$
View solution Problem 27
Solve each equation. $$ \log _{4}\left(x^{2}-3 x\right)=1 $$
View solution Problem 27
Write each as a logarithmic equation. $$ 5^{1 / 2}=\sqrt{5} $$
View solution