1.Use the laws of logarithms to express log2 (6) – log2 (3) + 2log2 (8)^1/2 as a single logarithm; then evaluate.
2. For logy = log(0.5x – 3) + log2, express y as a function of x.
X2
1. To express log2(6) - log2(3) + 2log2(8)^1/2 as a single logarithm, we can use the following logarithmic properties:
a.
For any positive numbers a and b, log(a) - log(b) = log(a/b).
Using this property, we can simplify the expression:
log2(6) - log2(3) = log2(6/3) = log2(2) = 1.
b.
For any positive number a and integer b, log(a^b) = b * log(a).
Using this property, we can simplify the expression:
2log2(8)^1/2 = 2(1/2 * log2(8)) = log2(8) = 3.
Now, our simplified expression becomes:
log2(6) - log2(3) + 2log2(8)^1/2 = 1 - 3 = -2.
So, the expression as a single logarithm is log2(1/4), and its value is -2.
2. To express y as a function of x in the equation logy = log(0.5x - 3) + log2, we can follow these steps:
a.
Combine the logarithms using the property:
log(a) + log(b) = log(a * b).
log(0.5x - 3) + log2 = log((0.5x - 3) * 2) = log(1x - 6) = log(x - 6).
b.
Since logy = log(x - 6), we can remove the logarithms by taking 10 raised to the power of both sides:
10^(logy) = 10^(log(x - 6)).
This simplifies to:
y = x - 6.
So, y is a function of x given by y = x - 6.