Can someone please double check my answers.
1. What pattern does the logarithmic function exhibit?
a. y = b+a(lnx)
b. y = a+b(lnx)
c. y = ax+b
d. y = axb
I think A and B both look right..I just went with B.
2. Let f(x) = log6x and g(x) = log1/2x. For all x>1, f(x)>g(x)
TRUE?
3. What is the value of the following logarithmic expression: y=log81
1.91?
4. A total of $20,000 is invested at an annual interest rate of 6%. No matter how many years this money is invested, what is the best investment plan to earn the most money in the end?
a. Compounded quarterly
b. Compounded monthly
c. Compounded daily
d. Compounded continuously
I am really unsure about this one. I would guess D
#1. No idea what "pattern" we are looking for. 'Splain?
#2. TRUE
#3. yes, if base 10
#4. D is correct. The more often compounded, the better return.
1. To determine the correct pattern for the logarithmic function, you need to understand the general form of a logarithmic equation. The form of a logarithmic function is written as y = a + b(ln x), where a and b are constants.
Looking at the given options:
a. y = b + a(ln x)
b. y = a + b(ln x)
c. y = ax + b
d. y = axb
From the given options, it seems that options A and B have the correct form of the logarithmic function, while options C and D do not have the ln (natural logarithm) term.
To determine the correct answer, you can compare the equation with the general form of a logarithmic function and see which option matches. In this case, option B (y = a + b(ln x)) is the correct answer.
2. To check if f(x) > g(x) holds for all x > 1 for the given functions f(x) = log6(x) and g(x) = log1/2(x), you can compare their values.
When x > 1:
f(x) = log6(x) will be positive, as the logarithm of a number greater than 1 is positive.
g(x) = log1/2(x) will be negative, as the logarithm of a number between 0 and 1 is negative.
Therefore, for all x > 1, f(x) > g(x) is TRUE.
3. The value of the logarithmic expression y = log81 can be found by considering the base of the logarithm. In this case, the base is not specified explicitly, which usually means that it is assumed to be 10.
To find the value of log81 with a base of 10, you need to determine to which power 10 must be raised to equal 81. In this case, 10^2 = 100, and 10^3 = 1000, so 10^2 < 81 < 10^3. Therefore, log81 is between 2 and 3.
The value you provided, 1.91, falls outside this range, so it is not the correct value for log81.
4. To determine the best investment plan to earn the most money in the end, you need to consider the compounding method that generates the highest compound interest.
Compound interest is calculated by multiplying the initial amount by the interest rate, and then adding that interest to the principal. The more frequently interest is compounded, the more frequently interest is added to the principal, resulting in higher overall returns.
From the given options:
a. Compounded quarterly
b. Compounded monthly
c. Compounded daily
d. Compounded continuously
The option that compounds interest continuously (option D) will result in the highest amount of money earned in the end. This is because continuous compounding allows for the maximum number of interest calculations throughout the investment period and maximizes the growth of the investment.