Math Problem Statement

^3\log \sqrt{8} + 2 \times ^3\log 5 - ^3\log 75 + ^3\log 18 - ^3\log 8 =

Solution

The expression provided in the image is:

3log8+2×3log53log75+3log183log8^3\log \sqrt{8} + 2 \times ^3\log 5 - ^3\log 75 + ^3\log 18 - ^3\log 8

Let's break this down step by step:

Step 1: Simplify the logarithmic terms.

Recall that: 3logab=b×3loga^3\log a^b = b \times ^3\log a 8=81/2\sqrt{8} = 8^{1/2}

Thus: 3log8=12×3log8^3\log \sqrt{8} = \frac{1}{2} \times ^3\log 8

So the expression becomes: 12×3log8+2×3log53log75+3log183log8\frac{1}{2} \times ^3\log 8 + 2 \times ^3\log 5 - ^3\log 75 + ^3\log 18 - ^3\log 8

Step 2: Combine the logarithmic terms.

Now, combine the terms involving 3log8^3\log 8:

12×3log83log8=12×3log8\frac{1}{2} \times ^3\log 8 - ^3\log 8 = -\frac{1}{2} \times ^3\log 8

Now, we have: 12×3log8+2×3log53log75+3log18-\frac{1}{2} \times ^3\log 8 + 2 \times ^3\log 5 - ^3\log 75 + ^3\log 18

Step 3: Use the properties of logarithms.

We know the logarithm of a product is the sum of the logarithms: 3loga3logb=3log(ab)^3\log a - ^3\log b = ^3\log \left(\frac{a}{b}\right) Thus: 12×3log8+2×3log53log75+3log18-\frac{1}{2} \times ^3\log 8 + 2 \times ^3\log 5 - ^3\log 75 + ^3\log 18 =12×3log8+2×3log5+3log(1875)= -\frac{1}{2} \times ^3\log 8 + 2 \times ^3\log 5 + ^3\log \left(\frac{18}{75}\right)

Step 4: Simplify the final expression.

1875=625=2×352\frac{18}{75} = \frac{6}{25} = \frac{2 \times 3}{5^2} So: 3log(1875)=3log(625)=3log(2×352)^3\log \left(\frac{18}{75}\right) = ^3\log \left(\frac{6}{25}\right) = ^3\log \left(\frac{2 \times 3}{5^2}\right)

Thus, the expression simplifies further to: 12×3log8+2×3log5+3log(2×325)-\frac{1}{2} \times ^3\log 8 + 2 \times ^3\log 5 + ^3\log \left(\frac{2 \times 3}{25}\right)

And since 125\frac{1}{25} can be expressed as 525^{-2}: =12×3log8+2×3log5+3log(2×3)2×3log5= -\frac{1}{2} \times ^3\log 8 + 2 \times ^3\log 5 + ^3\log (2 \times 3) - 2 \times ^3\log 5 =12×3log8+3log6= -\frac{1}{2} \times ^3\log 8 + ^3\log 6

Given the nature of this logarithm, the expression cannot be simplified further using basic algebraic techniques.

Final Answer: 12×3log8+3log6-\frac{1}{2} \times ^3\log 8 + ^3\log 6

Would you like more details on any step or do you have any further questions?


Relative Questions:

  1. How can you simplify logarithms with different bases?
  2. What is the significance of the properties of logarithms in simplifying expressions?
  3. How would the expression change if the logarithm base were different?
  4. What are some common logarithm identities used in simplifications?
  5. How can logarithms be applied to real-world problems, such as in finance or engineering?

Tip:

Always check if a logarithmic expression can be further simplified by using properties such as the product, quotient, or power rules before calculating.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Algebraic Simplification

Formulas

^b\log a^c = c \times ^b\log a
^b\log a - ^b\log b = ^b\log \left(\frac{a}{b}\right)

Theorems

Logarithm Properties

Suitable Grade Level

Grades 10-12