Consider two distinct positive real numbers $m, n$, with $m > n$. Let $x = n^{\log_{10}(m)}$ and $y = m^{\log_{10}(n)}$. The relation between $x$ and $y$ is _______.
Problem Analysis: We need to determine the relationship between two expressions, $x = n^{\log_{10}(m)}$ and $y = m^{\log_{10}(n)}$, given that $m$ and $n$ are distinct positive real numbers with $m > n$. This requires understanding properties of logarithms and exponents.
Comparing x and y using Logarithm Properties
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Analyze expression x: Let $x = n^{\log_{10}(m)}$. To simplify, we take the base-10 logarithm of both sides:
$ \log_{10}(x) = \log_{10}(n^{\log_{10}(m)}) $
Using the logarithm power rule, $\log(a^b) = b \log(a)$, we get:
$ \log_{10}(x) = (\log_{10}(m)) \cdot (\log_{10}(n)) $
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Analyze expression y: Let $y = m^{\log_{10}(n)}$. Similarly, take the base-10 logarithm of both sides:
$ \log_{10}(y) = \log_{10}(m^{\log_{10}(n)}) $
Applying the logarithm power rule:
$ \log_{10}(y) = (\log_{10}(n)) \cdot (\log_{10}(m)) $
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Compare logarithms: By comparing the results from Step 1 and Step 2:
$ \log_{10}(x) = (\log_{10}(m)) \cdot (\log_{10}(n)) $
$ \log_{10}(y) = (\log_{10}(n)) \cdot (\log_{10}(m)) $
Since the order of multiplication does not matter (i.e., $a \cdot b = b \cdot a$), we have:
$ \log_{10}(x) = \log_{10}(y) $
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Conclude the relation: The logarithm function is a one-to-one function. This means if $\log_{10}(x) = \log_{10}(y)$, then $x$ must be equal to $y$. The conditions that $m, n$ are positive distinct real numbers ensure that these expressions are well-defined.
Therefore, the relation is $x = y$.
Important Questions from Logarithms
For positive non-zero real variables $p$ and $q$, if
$\log (p^2 + q^2) = \log p + \log q + 2 \log 3$,
then, the value of $\frac{p^4+q^4}{p^2q^2}$ isFor a real number $x > 1$,
$\frac{1}{\log_2 x} + \frac{1}{\log_3 x} + \frac{1}{\log_4 x} = 1$
The value of $x$ isA petrified wood fossil was discovered with 8 g of $^{14}C$. The decay of $^{14}C$ over time is given by:
$N_T = N_0 e^{-0.0001216T}$
If the half-life of $^{14}C$ is 5700 years, and the fossil initially had 32 g of $^{14}C$, the age of the fossil in years is ______.
- For integers $a$, $b$ and $c$, what would be the minimum and maximum values respectively of $a + b + c$ if $\log |a| + \log |b| + \log |c| = 0$?
- The value of the expression $\frac{1}{1+\log_u vw} + \frac{1}{1+\log_v wu} + \frac{1}{1+\log_w uv}$ is
