1.
The equation 2 ^{x} = 9 can be entered in the calculator in which form?
Correct Answer
A. (log 9) / (log 2)
Explanation
To solve the equation 2x = 9, we need to isolate x. One way to do this is by using logarithms. Taking the logarithm of both sides of the equation, we get log(2x) = log(9). By the logarithmic property, we can bring down the exponent, so x * log(2) = log(9). To isolate x, we divide both sides of the equation by log(2), giving us x = log(9) / log(2). Therefore, the correct form to enter the equation 2x = 9 into the calculator is (log 9) / (log 2).
2.
A logarithmic function is the inverse of which function?
Correct Answer
D. Exponential
Explanation
A logarithmic function is the inverse of an exponential function. This means that if we have an exponential function y = a^x, the logarithmic function that undoes this operation is y = log base a of x. The logarithmic function "undoes" the exponential function by finding the power to which the base must be raised to obtain a certain value. Therefore, the correct answer is exponential.
3.
Condense using log properties, then simplify
Correct Answer
B. 3
Explanation
The given expression can be simplified by using the property of logarithms that states log(a) + log(b) = log(a * b). By applying this property, we can condense the expression into log(2 * 3 * 4 * 8). Further simplification gives us log(192), which is equal to 3.
4.
Expand: log_{2}8mn
Correct Answer
A. 3 + log_{2}m + log_{2}n
Explanation
The given expression is log28mn. Using the logarithmic property log(ab) = log(a) + log(b), we can rewrite the expression as log28 + log2m + log2n. Therefore, the correct answer is 3 + log2m + log2n.
5.
Write as a single logarithm: 3logm - 2logn - 0.5logp
Correct Answer
C. Log (m^{3}/(n^{2}p^{0.5})
Explanation
The given expression can be simplified by using the properties of logarithms. By applying the power rule of logarithms, we can rewrite the expression as log(m^3) - log(n^2) - log(p^0.5). Then, using the quotient rule of logarithms, we can combine the terms to get log(m^3/(n^2*p^0.5)). Therefore, the correct answer is log(m^3/(n^2*p^0.5)).
6.
The following expression: 3 - log_{3}n is the correct expansion of : log_{3}(27/n)
Correct Answer
A. True
Explanation
The expression 3 - log3n is the correct expansion of log3(27/n) because when we apply the logarithmic property loga(b/c) = loga(b) - loga(c), we can rewrite log3(27/n) as log3(27) - log3(n), which simplifies to 3 - log3n.
7.
Logarithmic functions always have ________ asymptotes?
Correct Answer
Vertical
Explanation
Logarithmic functions always have vertical asymptotes. This is because the domain of a logarithmic function is restricted to positive real numbers. As the input approaches zero, the output of the logarithmic function approaches negative infinity. Therefore, the graph of a logarithmic function approaches a vertical line, which is the vertical asymptote, as the input approaches zero.
8.
Describe the following expression as a single logarithm: (1/2)logm + (1/3) logn
Correct Answer
B. Log (square root of m, times, the cube root of n)
Explanation
The given expression (1/2)logm + (1/3)logn can be simplified by using the property of logarithms that states loga + logb = log(ab). Applying this property, we can rewrite the expression as log(sqrt(m)) + log(cbrt(n)). Since the logarithm of a product is equal to the sum of the logarithms of its factors, this can be further simplified to log(sqrt(m) * cbrt(n)). Therefore, the correct answer is "log (square root of m, times, the cube root of n)".
9.
What is the parent function of : f(x)=log_{2}(x-1)+3
Correct Answer
A. F(x)=log_{2}x
Explanation
The correct answer is f(x)=log2x. This is because the given function f(x)=log2(x-1)+3 is a transformation of the parent function f(x)=log2x. The +3 in the given function represents a vertical shift upwards by 3 units, but the base function remains the same. Therefore, the parent function is f(x)=log2x.
10.
Use the Change of Base formula to evaluate the following: log_{3}20
Correct Answer
D. 2.727
Explanation
The correct answer is 2.727. To evaluate log320, we can use the change of base formula which states that log base a of b can be calculated as log base c of b divided by log base c of a. In this case, we can use the natural logarithm (log base e) as the base c. So, log320 can be calculated as log base e of 20 divided by log base e of 3. Evaluating this expression gives us approximately 2.727.