anemone said:Which of the following two is greater?
$\log_9 25$ and $\log_4 9$
anemone said:Very good, kaliprasad and thanks for participating!
My solution:
$\log_9 25<\log_9 26<\dfrac{\log_4 26}{\log_4 9}<\dfrac{\log_4 26}{\log_4 8}=\dfrac{\log_4 (26)^2}{3}=\dfrac{\log_4 676}{3}<\dfrac{\log_4 729}{3}=\log_4 9$
The two expressions are logarithms with a base of 9 and 4, respectively. They represent the power to which the base must be raised to equal the given number. In this case, $\log_9 25$ is asking "what power of 9 equals 25?" while $\log_4 9$ is asking "what power of 4 equals 9?"
To compare logarithms with different bases, we can use the change of base formula: $\log_a b = \frac{\log_c b}{\log_c a}$. This allows us to rewrite the expressions as $\frac{\log 25}{\log 9}$ and $\frac{\log 9}{\log 4}$, making it easier to compare.
To determine which is greater, we can evaluate the expressions using a calculator. In this case, $\log_9 25 \approx 1.5563$ and $\log_4 9 \approx 1.5$, so $\log_9 25$ is greater.
Yes, we can simplify both expressions using the properties of logarithms. $\log_9 25$ can be rewritten as $\frac{\log 25}{\log 9}$, and since $\log 25 = 2$ and $\log 9 = \frac{2}{3}$, we get $\frac{2}{\frac{2}{3}} = 3$. Similarly, $\log_4 9$ can be rewritten as $\frac{\log 9}{\log 4}$, and since $\log 9 = \frac{1}{2}$ and $\log 4 = \frac{1}{2}$, we get $\frac{\frac{1}{2}}{\frac{1}{2}} = 1$. Therefore, $\log_9 25 = 3$ and $\log_4 9 = 1$.
Comparing logarithms with different bases helps us understand the relationships between different powers and how they relate to each other. It also allows us to simplify expressions and solve equations involving logarithms more easily.