Evaluate (log2 3)/(log2 9)
Using the change of base formula, we have:
(log2 3)/(log2 9) = log3/log9
Applying the change of base formula again, we have:
log3/log9 = log3/(log3 3^2) = log3/2
Therefore,
(log2 3)/(log2 9) = log3/2 ≈ 0.585
1/2
To evaluate the expression (log₂ 3) / (log₂ 9), we can use the property of logarithms that states log base a of b divided by log base a of c is equal to log base c of b.
In other words, (log base a of b) / (log base a of c) = log base c of b.
Applying this property to our expression, we have:
(log₂ 3) / (log₂ 9) = log₉ 3
To further simplify this expression, we need to use the change-of-base formula for logarithms. The change-of-base formula allows us to convert the logarithm with any base to a logarithm with a different base.
The change-of-base formula states that log base a of b is equal to log base c of b divided by log base c of a.
Using the change-of-base formula, we can express log₉ 3 as:
log₉ 3 = log₂ 3 / log₂ 9
Now, we need to evaluate log₂ 3 and log₂ 9 separately.
To find log₂ 3, we need to determine to which power we need to raise 2 to get 3. Let's call the exponent x:
2^x = 3
Taking the logarithm base 2 of both sides, we have:
log₂ (2^x) = log₂ 3
x = log₂ 3
Therefore, log₂ 3 = x.
Similarly, to find log₂ 9, we need to determine to which power we need to raise 2 to get 9. Let's call the exponent y:
2^y = 9
Taking the logarithm base 2 of both sides, we have:
log₂ (2^y) = log₂ 9
y = log₂ 9
Therefore, log₂ 9 = y.
Now, substituting the values of log₂ 3 and log₂ 9 back into the expression we're evaluating, we have:
(log₂ 3) / (log₂ 9) = x / y
So, to evaluate (log₂ 3) / (log₂ 9), we need to find the values of x and y, which are log₂ 3 and log₂ 9, respectively.