Math Problem Statement

Make it to a simple log: log2(2y^2)

Solution

To simplify log2(2y2)\log_2(2y^2), let's break it down using logarithmic properties.

Step-by-Step Solution:

  1. We start with: log2(2y2)\log_2(2y^2)

  2. Using the product rule of logarithms, logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y): log2(2y2)=log2(2)+log2(y2)\log_2(2y^2) = \log_2(2) + \log_2(y^2)

  3. Now, we know that log2(2)=1\log_2(2) = 1 because 21=22^1 = 2: =1+log2(y2)= 1 + \log_2(y^2)

  4. Next, apply the power rule of logarithms, logb(xk)=klogb(x)\log_b(x^k) = k \cdot \log_b(x): =1+2log2(y)= 1 + 2\log_2(y)

Final Answer:

log2(2y2)=1+2log2(y)\log_2(2y^2) = 1 + 2\log_2(y)

Would you like further clarification on any step?


Here are some related questions you might find useful:

  1. How do the properties of logarithms simplify expressions with multiple terms?
  2. What are some common mistakes to avoid when simplifying logarithmic expressions?
  3. How would you simplify an expression like log3(9x3)\log_3(9x^3)?
  4. What are the applications of logarithmic simplifications in calculus?
  5. How do base changes affect logarithmic simplifications?

Tip: Always remember the three core properties of logarithms—product, quotient, and power rules—to simplify complex expressions efficiently.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Logarithmic Simplification
Algebra

Formulas

Product rule of logarithms: log_b(xy) = log_b(x) + log_b(y)
Power rule of logarithms: log_b(x^k) = k * log_b(x)

Theorems

Properties of Logarithms

Suitable Grade Level

Grades 9-11