Math Problem Statement

Simplify the expression: 2 log 8 + 6 log 3 - log 12 - log 5.

Solution

The expression in the image can be interpreted as follows:

2log8+6log3log12log52 \log 8 + 6 \log 3 - \log 12 - \log 5

Let's simplify this expression step by step using logarithmic properties. We'll use the following logarithmic rules:

  1. loga(xn)=nlogax\log_a(x^n) = n \log_a x
  2. logax+logay=loga(xy)\log_a x + \log_a y = \log_a (x \cdot y)
  3. logaxlogay=loga(xy)\log_a x - \log_a y = \log_a \left(\frac{x}{y}\right)

Step 1: Apply the first property to the terms with coefficients

2log8=log(82)=log642 \log 8 = \log (8^2) = \log 64 [ 6 \log 3 = \log (3^6) = \log 729

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Math Problem Analysis

Mathematical Concepts

Logarithms
Algebra
Simplification of Expressions

Formulas

log_a(x^n) = n log_a x
log_a x + log_a y = log_a (x * y)
log_a x - log_a y = log_a (x / y)

Theorems

Logarithmic Properties

Suitable Grade Level

Grades 10-12