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Magazine Chapter 13: Problem 61
\(\text {Solve the given problems.}\) If \(\log _{b} x=2\) and \(\log _{b} y=3,\) find \(\log _{b} \sqrt{x^{2} y^{4}}\)
Short Answer
Expert verified
\(\log_b \sqrt{x^2 y^4} = 8\)
Step by step solution
01
Express the Logarithms in Terms of Given Values
We are given \(\log_b x = 2\) and \(\log_b y = 3\). We need to find \(\log_b \sqrt{x^2 y^4}\). Start by expressing the logarithm we need to find using the given values:\[\log_b \sqrt{x^2 y^4} = \log_b (x^2 y^4)^{1/2}\]
02
Simplify the Expression Using Power Rules
Apply the power rule of logarithms, which states \( \log_b (a^n) = n \times \log_b(a) \):\[\log_b (x^2 y^4)^{1/2} = \frac{1}{2} \cdot \log_b (x^2 y^4)\]Now apply the product rule of logarithms, \( \log_b (ab) = \log_b a + \log_b b \):\[\log_b (x^2 y^4) = \log_b x^2 + \log_b y^4\]
03
Substitute and Simplify Each Logarithm
Using the power rule again, replace each term with given values:\[\log_b x^2 = 2 \times \log_b x = 2 \times 2 = 4\]\[\log_b y^4 = 4 \times \log_b y = 4 \times 3 = 12\]Combine these results:\[\log_b (x^2 y^4) = 4 + 12 = 16\]
04
Complete the Calculation
Substitute back into the expression from Step 2:\[\frac{1}{2} \cdot 16 = 8\]
05
Conclude with the Final Result
The logarithm \(\log_b \sqrt{x^2 y^4}\) simplifies to 8. Therefore, the final solution is:\[\log_b \sqrt{x^2 y^4} = 8\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Logarithm rules
Logarithms are a very powerful tool in mathematics, especially when dealing with exponents and algebraic expressions. They help us solve complex equations by converting multiplication into addition, a process that simplifies many problems. The main rules to keep in mind are:
- The **product rule**: For any base \( b \), the logarithm of a product \( \log_b (xy) = \log_b x + \log_b y \).
- The **quotient rule**: Similarly, the logarithm of a division \( \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y \).
- The **power rule**: For a power \( a^n \), the logarithm can be expressed as \( \log_b (a^n) = n \cdot \log_b(a) \).
Exponents
Exponents are a way to express repeated multiplication of a number by itself. They are denoted by a small number, called the exponent, placed to the upper right of a base number. For example, \( x^2 \) means \( x \times x \). Dealing with exponents regularly involves some key rules:
- The **product of powers rule**: \( a^m \times a^n = a^{m+n} \) combines bases with the same exponent by adding their powers.
- The **power of a power rule**: \( (a^m)^n = a^{m\times n} \) follows the multiplication of the exponents.
- The **power of a product rule**: \( (ab)^n = a^n \times b^n \), helps in distributing the exponent over multiplication.
Algebraic expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They help in expressing real-world problems in terms of equations and inequalities. Understanding how to manipulate these expressions is key to solving them effectively. Here are some foundational concepts to keep in mind:
- A **term** is a single mathematical entity; it can be a number, a variable, or numbers and variables multiplied together.
- A **factor** of a term is one of its components, such as \( x \) in \( 3x \).
- An **expression** is a group of terms combined using addition, subtraction, multiplications, and/or division operators.
