Math Problem Statement

Simplify log₃ 81 − log₅ 125 + log√2 8.

Solution

To simplify the given expression, let's work through it step by step:

The expression is:

log381log5125+log28.\log_3 81 - \log_5 125 + \log_{\sqrt{2}} 8.

Step 1: Simplify log381\log_3 81

We know that 81=3481 = 3^4. Using the logarithmic property logaab=b\log_a a^b = b, we have: log381=log3(34)=4.\log_3 81 = \log_3 (3^4) = 4.

Step 2: Simplify log5125\log_5 125

We know that 125=53125 = 5^3. Similarly, using the same property: log5125=log5(53)=3.\log_5 125 = \log_5 (5^3) = 3.

Step 3: Simplify log28\log_{\sqrt{2}} 8

We rewrite 88 as (2)6(\sqrt{2})^6 because 8=238 = 2^3 and 2=21/2\sqrt{2} = 2^{1/2}. Thus: 8=(21/2)6=(2)6.8 = (2^{1/2})^6 = (\sqrt{2})^6. Now, using logaab=b\log_a a^b = b, we have: log28=log2(2)6=6.\log_{\sqrt{2}} 8 = \log_{\sqrt{2}} (\sqrt{2})^6 = 6.

Step 4: Combine the results

Substitute the simplified values back into the expression: log381log5125+log28=43+6.\log_3 81 - \log_5 125 + \log_{\sqrt{2}} 8 = 4 - 3 + 6. Simplify further: 43+6=7.4 - 3 + 6 = 7.

Final Answer:

7\boxed{7}

Let me know if you would like further clarification or additional examples to deepen your understanding!

Here are 5 related questions to explore further:

  1. How do logarithmic properties simplify calculations?
  2. What are some other common bases where logarithmic expressions are used?
  3. How can you convert between logarithms of different bases?
  4. How do you evaluate logab\log_a b when bb is not a power of aa?
  5. What is the difference between natural logarithms and common logarithms?

Tip: Always remember to rewrite terms into a base or exponent form that simplifies calculations when working with logarithms!

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Math Problem Analysis

Mathematical Concepts

Logarithms
Exponents
Logarithmic Properties

Formulas

logₐ(aᵇ) = b
logₐ(b) − logₐ(c) = logₐ(b/c)
logₐ(b) + logₐ(c) = logₐ(bc)

Theorems

Logarithmic Exponentiation Rule
Change of Base Theorem

Suitable Grade Level

Grades 10-12