Math Problem Statement

³log54+³log18-³log 12

Solution

To simplify the expression:

3log54+3log183log12^3\log 54 + ^3\log 18 - ^3\log 12

We use the logarithmic properties:

  1. Logarithm of a product: alog(xy)=alogx+alogy^a\log(xy) = ^a\log x + ^a\log y
  2. Logarithm of a quotient: alog(xy)=alogxalogy^a\log\left(\frac{x}{y}\right) = ^a\log x - ^a\log y

Let’s simplify step by step.

  1. Combine the first two logarithms using the product rule: 3log54+3log18=3log(54×18)=3log972^3\log 54 + ^3\log 18 = ^3\log(54 \times 18) = ^3\log 972

  2. Now subtract 3log12^3\log 12: 3log9723log12=3log(97212)=3log81^3\log 972 - ^3\log 12 = ^3\log\left(\frac{972}{12}\right) = ^3\log 81

  3. Since 81=3481 = 3^4, the logarithm becomes: 3log81=3log34=4^3\log 81 = ^3\log 3^4 = 4

Thus, the value of the given expression is:

44

Would you like more details on any specific step?

Here are some related questions:

  1. How does the change of base formula apply to logarithms?
  2. Can you explain other properties of logarithms, like the power rule?
  3. How would the result change if the logarithm base was different?
  4. What are the common applications of logarithms in real-life problems?
  5. How can we express logarithms in terms of natural logs?

Tip: Always remember to simplify logarithmic expressions step by step using product and quotient rules first!

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Math Problem Analysis

Mathematical Concepts

Logarithms
Logarithmic Properties
Simplification

Formulas

Logarithm of a product: log_a(xy) = log_a x + log_a y
Logarithm of a quotient: log_a(x/y) = log_a x - log_a y

Theorems

Logarithmic Properties

Suitable Grade Level

Grades 10-12