i need help with these two homework problems
Use the Laws of Logarithms to combine the expression into a single logarithm
log2 5 − 5 log2 x + 1/2 log2(x + 1)
Solve the logarithmic equation for x
log2(x + 2) + log2(x − 1) = 2
log2 [ 5*sqrt(x+1)/x^5 ]
log2 (x^2+x-2) = 2
x^2 + x - 2 = 2^2 = 4
x^2 +x -6 = 0
(x-2)(x+3) = 0
x = 2 or -3
Sure! I can help you with both of these problems.
Problem 1:
To combine the expression into a single logarithm using the Laws of Logarithms, we need to apply the following properties:
1. log(a) + log(b) = log(ab)
2. log(a) - log(b) = log(a/b)
3. log(a^n) = n*log(a)
Given: log2 5 - 5 log2 x + 1/2 log2(x + 1)
First, let's apply the second property to combine the first two terms:
log2 5 - 5 log2 x = log2 5 - log2(x^5) = log2 (5 / x^5)
Now, let's use the first property to combine the result with the last term:
log2 (5 / x^5) + 1/2 log2(x + 1) = log2 [(5 / x^5) * √(x + 1)]
Therefore, the expression can be simplified to a single logarithm as log2 [(5 / x^5) * √(x + 1)].
Problem 2:
To solve the logarithmic equation log2(x + 2) + log2(x - 1) = 2, we can use the property:
1. log(a) + log(b) = log(ab)
Given: log2(x + 2) + log2(x - 1) = 2
Using the property mentioned above, we can rewrite the equation as a single logarithm:
log2[(x + 2) * (x - 1)] = 2
Now, let's convert the equation into exponential form:
2^2 = (x + 2) * (x - 1)
Simplifying the right side, we have:
4 = x^2 + x - 2
Rearranging the equation to the standard form, we get:
x^2 + x - 6 = 0
Now, we can factorize the quadratic equation:
(x + 3)(x - 2) = 0
Setting each factor equal to zero and solving for x, we have:
x + 3 = 0 --> x = -3
x - 2 = 0 --> x = 2
So, the solutions to the logarithmic equation are x = -3 and x = 2.