Use the properties of logarithms to find the value the following expression.
log3 9^4
the answer options are
a. 2
b. 4
c. 6
d. 8
e. 10
I got 4?
log3 9^4
= 4 log3 9
= 4(2)
= 8
or , just think it through
9^4 = 3^8
so log3 3^8 must be 8
To find the value of the expression log3 9^4, we can use the properties of logarithms.
One useful property is that loga b^c = c * loga b.
Applying this property to the given expression, we have:
log3 9^4 = 4 * log3 9
Another logarithmic property that can be useful is that loga a = 1. In this case, we can rewrite the base 3 logarithm as a base 9 logarithm using the fact that 3^2 = 9:
log3 9 = log9 9^(2/2) = log9 (3^2)
Now, we can apply the logarithmic property loga b^c = c * loga b again:
4 * log3 9 = 4 * log9 (3^2)
Since log9 (3^2) = 2, we can simplify the expression further:
4 * log9 (3^2) = 4 * 2 = 8
According to the calculation, the value of the expression log3 9^4 is 8. Therefore, the answer is option d: 8.