Evaluate log9 162 - log9 (2/3)
Using the logarithmic identity log a - log b = log (a/b), we can simplify the expression:
log9 162 - log9 (2/3) = log9 (162/(2/3))
= log9 (162 * 3/2)
= log9 243
We can also rewrite 243 as a power of 9: 243 = 9^2.5
Therefore, log9 243 = 2.5.
So,
log9 162 - log9 (2/3) = 2.5.
2 1/2
2 1/2 can be written as an improper fraction as follows:
2 1/2 = 5/2
Therefore, 2 1/2 is equal to 5/2.
To evaluate the expression log9 162 - log9 (2/3), we can use the logarithmic properties.
First, let's simplify each logarithm individually. Recall the logarithmic identity logb a - logb c = logb (a/c).
For the first term, log9 162, we can rewrite 162 as a power of 9 since we're working with a base of 9. So, 162 = 9^2 * 2 = 9^2 * (2/9).
Now, we can rewrite log9 162 as log9 (9^2 * (2/9)), which can be further simplified using the above logarithmic identity to log9 (9^2 * (2/9)) = log9 (9^2/9) = log9 (9/1) = log9 9.
Since log9 9 is equivalent to log9 9^1, which is just 1, the first term simplifies to 1.
Moving on to the second term, log9 (2/3), we apply the same logic. We can write 2/3 as a power of 9, so 2/3 = (2/9) * 9.
Then, we simplify log9 (2/3) as log9 ((2/9) * 9), which becomes log9 (2/9) + log9 9.
We know that log9 9 is equal to 1, so the second term simplifies to log9 (2/9) + 1.
Finally, substituting the simplified terms back into the original expression, we have:
log9 162 - log9 (2/3) = 1 - (log9 (2/9) + 1).
As the "1" in the expression cancels out with "-1", we are left with:
log9 162 - log9 (2/3) = -log9 (2/9).
So, the expression simplifies to -log9 (2/9).