for what values of x is log_0.75(x+3)>log_0.5(x+3)
A.-2<x<∞
B.-3<x<-2
C.4<x<∞
D.3<x<4
which of the following logarithmic expressions are equivalent to In sqr xy+In(x/e)? choose three correct options
A.In(sqr x^3y/e)
B.In(x^2y/2e)
C.1/2In x^2y-e
D.In(x sqr xy)-1
E. 3/2In x+1/2In y-1
solve log_81 32x-log_81(x-3)=3/4 for x
A. x=24/31
B.x=-81/5
C.x=729 3sqr3/243 3sqr3-32
D. no solution
state the domain, range, intercept(s), and asymptote(s) of y=-2^-x+2+1
To solve the inequality log₀.₇₅(x+3) > log₀.₅(x+3):
First, define a new variable y = x + 3. The inequality becomes:
log₀.₇₅(y) > log₀.₅(y)
Now, since the base and the argument of the logarithm are the same, the inequality simplifies to:
y > y
This inequality is true for all y. Therefore, substituting back x + 3 for y, we get:
x + 3 > x + 3
This equation is also true for all x. Therefore, the solution is -2 < x < ∞.
Answer: A. -2 < x < ∞
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To determine which logarithmic expressions are equivalent to In(√(xy)) + In(x/e), we simplify the expression first:
In(√(xy)) + In(x/e)
= In(√(xy)(x/e))
= In(√(x³y/e))
= In(√(x²y/e) * √(x/e))
= In((x²y/e)^(1/2) * (x/e)^(1/2))
= In(x²y/e * x/e)
= In(x²y/2e)
Therefore, the equivalent expressions are:
A. In(x²y/2e)
C. 1/2In(x²y-e)
D. In(x(√(xy))-1)
Answer: A. In(x²y/2e), C. 1/2In(x²y-e), D. In(x(√(xy))-1)
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To solve log₈₁(32x) - log₈₁(x-3) = 3/4:
Using the properties of logarithms, we can rewrite the equation as:
log₈₁(32x / (x-3)) = 3/4
Now, we can convert the equation to exponential form:
81^(3/4) = 32x / (x-3)
3³ = 32x / (x-3)
27 = 32x / (x-3)
Cross-multiply:
27(x-3) = 32x
27x - 81 = 32x
-81 = 5x
x = -81/5
Therefore, the solution is:
B. x = -81/5
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For the equation y = -2^(-x+2) + 1:
Domain: Since we have a negative exponent, the base 2 cannot equal zero. Therefore, the domain of the function is all real numbers.
Range: The range of the function y = -2^(-x+2) + 1 is y < 3.
Intercept: To find the y-intercept, set x to 0:
y = -2^(-0+2) + 1
y = -2² + 1
y = -4 + 1
y = -3
Therefore, the y-intercept is -3.
Asymptote: There is a horizontal asymptote at y = 1, as x approaches positive or negative infinity.
Domain: All real numbers
Range: y < 3
Intercept: y-intercept is -3
Asymptote: Horizontal asymptote at y = 1.
Answer: Domain: All real numbers, Range: y < 3, Intercept: y-intercept is -3, Asymptote: Horizontal asymptote at y = 1.