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已知三个互不相等的正数

a, b, c

满足 $a=\mathrm{e}^{2}, b=\log _{2} 3+\log _{9} 6, c=\log _{...

Mar 8, 2024

已知三个互不相等的正数

a, b, c

满足

a=\mathrm{e}^{2}, b=\log _{2} 3+\log _{9} 6, c=\log _{\sqrt{5}}\left(2^{a}+1\right)

，（其中

\mathrm{c}=2.71828 \cdots

是一个无理数），则

a, b, c

的大小关系为 ( ) A. a<b<c B. a<c<b C. \mathrm{c}<\mathrm{a}<\mathrm{b} D. c<b<a

Generated Graph

Solution by Steps

step 1

Solve for

a

using the equation

a = e^2

step 2

Calculate

b

using the logarithmic properties:

b = \log_2(3) + \log_9(6)

. Convert

\log_9(6)

to base 2 using change of base formula:

\log_9(6) = \frac{\log_2(6)}{\log_2(9)}

step 3

Simplify

b

by calculating the logarithms:

b = \log_2(3) + \frac{\log_2(6)}{\log_2(9)}

step 4

Solve for

c

using the given logarithmic equation:

c = \log_{\sqrt{5}}(2^a + 1)

step 5

Substitute

a = e^2

into the equation for

c

and simplify:

c = \frac{1}{2} \log_5(2^{e^2} + 1)

step 6

Compare the values of

a

b

, and

c

to determine their relationship

Answer

a < c < b

Key Concept

Comparing values of expressions involving exponents and logarithms

Explanation

The value of

a

is a constant since

e

is a known irrational number. The value of

b

involves logarithms with different bases, which can be simplified and compared. The value of

c

is dependent on

a

and involves a logarithm that can be simplified. By evaluating these expressions, we can compare their magnitudes to determine the relationship between

a

b

, and

c