Math Problem Statement

Calculate the value of the expression 4^log25 * 5^log36 - 2^log9 and select the correct answer from the given options.

Solution

To solve the given expression, we need to evaluate:

4log255log362log94^{\log 25} \cdot 5^{\log 36} - 2^{\log 9}

Let's break down each term step-by-step:

Step-by-Step Evaluation:

  1. Evaluate 4log254^{\log 25}: 4=22and25=524 = 2^2 \quad \text{and} \quad 25 = 5^2 So: 4log25=(22)log(52)=22log(52)=222log5=24log5=(2log5)44^{\log 25} = (2^2)^{\log (5^2)} = 2^{2 \cdot \log (5^2)} = 2^{2 \cdot 2 \log 5} = 2^{4 \log 5} = (2^{\log 5})^4 Since 2log5=52^{\log 5} = 5, we have: (2log5)4=54=625(2^{\log 5})^4 = 5^4 = 625

  2. Evaluate 5log365^{\log 36}: 36=62=(23)2=223236 = 6^2 = (2 \cdot 3)^2 = 2^2 \cdot 3^2 log36=log(2232)=2log2+2log3=2(log2+log3)\log 36 = \log (2^2 \cdot 3^2) = 2 \log 2 + 2 \log 3 = 2 (\log 2 + \log 3) Therefore: 5log36=52(log2+log3)=(5log2+log3)2=(5log25log3)2=(23)2=62=365^{\log 36} = 5^{2 (\log 2 + \log 3)} = (5^{\log 2 + \log 3})^2 = (5^{\log 2} \cdot 5^{\log 3})^2 = (2 \cdot 3)^2 = 6^2 = 36

  3. Multiply the two results: 4log255log36=625364^{\log 25} \cdot 5^{\log 36} = 625 \cdot 36 62536=22500625 \cdot 36 = 22500

  4. Evaluate 2log92^{\log 9}: 9=32    log9=log(32)=2log39 = 3^2 \implies \log 9 = \log (3^2) = 2 \log 3 So: 2log9=22log3=(2log3)22^{\log 9} = 2^{2 \log 3} = (2^{\log 3})^2 Since 2log3=32^{\log 3} = 3, we get: (2log3)2=32=9(2^{\log 3})^2 = 3^2 = 9

  5. Subtract the last term: 225009=2249122500 - 9 = 22491

Given the choices, it seems the problem may have intended different properties of logarithms or base conversions, but assuming the straightforward interpretation above:

22491\boxed{22491}

However, none of the provided options match this result. The provided answer choices might indicate a different expected approach, such as base manipulation or simplifications assumed in a different way. Double-check the problem context for further clues!

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Math Problem Analysis

Mathematical Concepts

Logarithms
Exponents
Properties of Logarithms
Simplification

Formulas

a^log(b) = b^(log(a))
log(a^b) = b * log(a)
Product and Power Rules of Exponents

Theorems

Change of Base Theorem
Exponentiation and Logarithm Rules

Suitable Grade Level

Grades 10-12