Evaluate the following expression:
\[ \frac{\log_{10} 5 \cdot \log_{10} 20 + (\log_{10} 2)^2}{5^{\log_{\sqrt{5}} 2} + 9^{\log_3 7} - 8^{\log_2 5}} \]
Answer (2)
The evaluated expression yields -\frac{1}{72}. This was obtained by simplifying the numerator to 1 and the denominator to -72. The final result is therefore -\frac{1}{72}.
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To evaluate the given expression:
5 l o g 5 2 + 9 l o g 3 7 − 8 l o g 2 5 lo g 10 5 ⋅ lo g 10 20 + ( lo g 10 2 ) 2
we need to break it down step by step.
First, let's simplify the numerator:
lo g 10 20 = lo g 10 ( 4 × 5 ) = lo g 10 4 + lo g 10 5 = 2 lo g 10 2 + lo g 10 5 using the properties of logarithms.
Substitute lo g 10 20 into the expression: lo g 10 5 ⋅ ( 2 lo g 10 2 + lo g 10 5 ) + ( lo g 10 2 ) 2
Expand the expression: 2 lo g 10 5 ⋅ lo g 10 2 + ( lo g 10 5 ) 2 + ( lo g 10 2 ) 2
Now, let's simplify the denominator:
5 l o g 5 2 = 2 1 = 2 due to the identity a l o g a b = b .
9 l o g 3 7 = ( 3 2 ) l o g 3 7 = 7 2 = 49 .
8 l o g 2 5 = ( 2 3 ) l o g 2 5 = 5 3 = 125 .
Substitute these simplifications into the expression:
2 + 49 − 125 2 lo g 10 5 ⋅ lo g 10 2 + ( lo g 10 5 ) 2 + ( lo g 10 2 ) 2
The denominator becomes 51 − 125 = − 74 .
So, the complete expression is:
− 74 2 lo g 10 5 ⋅ lo g 10 2 + ( lo g 10 5 ) 2 + ( lo g 10 2 ) 2
While we have simplified it down, exact numerical evaluation could depend on a calculator for exact decimal values for the logarithms to get an approximate decimal number.
This is a high-level manipulation of logarithms, typically taught in upper High School mathematics or entry-level college mathematics.
