Math Problem Statement

2log 50 + 2 Log 8 - 2 Log 100

Solution

We can simplify the expression 2log50+2log82log1002\log 50 + 2\log 8 - 2\log 100 using logarithmic properties. Here's the step-by-step solution:

Step 1: Apply the logarithm power rule

The logarithm power rule states that alogb=logbaa\log b = \log b^a. Let's apply this to each term:

2log50=log5022\log 50 = \log 50^2 2log8=log822\log 8 = \log 8^2 2log100=log10022\log 100 = \log 100^2

So the expression becomes:

log502+log82log1002\log 50^2 + \log 8^2 - \log 100^2

Step 2: Simplify the exponents

Now calculate the squares:

log502=log2500\log 50^2 = \log 2500 log82=log64\log 8^2 = \log 64 log1002=log10000\log 100^2 = \log 10000

Thus, the expression is now:

log2500+log64log10000\log 2500 + \log 64 - \log 10000

Step 3: Apply the logarithm addition and subtraction rules

The logarithm addition rule states loga+logb=log(a×b)\log a + \log b = \log (a \times b), and the subtraction rule states logalogb=log(ab)\log a - \log b = \log \left(\frac{a}{b}\right).

First, combine the terms using the addition rule:

log(2500×64)log10000\log (2500 \times 64) - \log 10000

Step 4: Perform the multiplication

Calculate 2500×642500 \times 64:

2500×64=1600002500 \times 64 = 160000

So the expression becomes:

log160000log10000\log 160000 - \log 10000

Step 5: Apply the subtraction rule

Finally, apply the subtraction rule:

log(16000010000)\log \left(\frac{160000}{10000}\right)

Step 6: Simplify the division

Simplify the fraction:

16000010000=16\frac{160000}{10000} = 16

So the final simplified expression is:

log16\log 16

Final Answer:

The expression simplifies to log16\log 16.

Would you like more details on any part of this process?


Relative Questions:

  1. How do you use the change of base formula for logarithms?
  2. Can you explain the logarithm product rule in detail?
  3. How do logarithm properties apply to solving logarithmic equations?
  4. What are the common logarithms and their typical applications?
  5. How would you expand or condense more complex logarithmic expressions?

Tip: Always remember that logarithms turn multiplication into addition and division into subtraction, which is handy when simplifying expressions.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Logarithmic Properties
Exponentiation

Formulas

Logarithm Power Rule: a log b = log b^a
Logarithm Addition Rule: log a + log b = log (a * b)
Logarithm Subtraction Rule: log a - log b = log (a / b)

Theorems

Properties of Logarithms

Suitable Grade Level

Grades 9-11