Precalculus: Exponential and Logarithmic Functions Worksheet

Question 1

Evaluate f(x) = 3(2)^x for x = -2, x = 0, and x = 3.

Accepted Answer

The values are f(-2) = 3/4, f(0) = 3, and f(3) = 24 because 2^-2 = 1/4, 2^0 = 1, and 2^3 = 8.

Question 2

Rewrite each equation in the other form: log_5(125) = 3 and 10^-2 = 0.01.

Accepted Answer

The equation log_5(125) = 3 can be written as 5^3 = 125. The equation 10^-2 = 0.01 can be written as log_10(0.01) = -2.

Question 3

Solve for x: 2^(x + 1) = 16.

Accepted Answer

Since 16 = 2^4, the equation becomes 2^(x + 1) = 2^4. Therefore, x + 1 = 4 and x = 3.

Question 4

Solve for t: 7e^(0.4t) = 35.

Accepted Answer

Divide both sides by 7 to get e^(0.4t) = 5. Taking the natural logarithm gives 0.4t = ln(5), so t = ln(5)/0.4, which is approximately 4.024.

Question 5

For the function y = log_2(x - 3) + 1, identify the domain, range, vertical asymptote, and the transformation from y = log_2(x).

Accepted Answer

The domain is x > 3, the range is all real numbers, and the vertical asymptote is x = 3. The graph of y = log_2(x) is shifted right 3 units and up 1 unit.

Question 6

Expand the expression using logarithm properties: ln(5x^3/sqrt(y)), where x and y are positive.

Accepted Answer

The expanded expression is ln(5) + 3ln(x) - 1/2ln(y). This uses the quotient rule, product rule, and power rule for logarithms.

Question 7

Condense the expression into a single logarithm: 2log(x) - (1/3)log(y) + log(4), where x and y are positive.

Accepted Answer

The condensed expression is log(4x^2/y^(1/3)). The coefficients become exponents, addition becomes multiplication, and subtraction becomes division.

Question 8

Solve for x and check for extraneous solutions: log_3(x - 2) + log_3(x + 2) = 2.

Accepted Answer

Combine the logarithms to get log_3((x - 2)(x + 2)) = 2, so x^2 - 4 = 9. Then x^2 = 13, so x = plus or minus sqrt(13). The domain requires x > 2, so the solution is x = sqrt(13).

Question 9

Find the inverse of f(x) = 4e^(x - 1) - 6.

Accepted Answer

Start with y = 4e^(x - 1) - 6. Then y + 6 = 4e^(x - 1), so (y + 6)/4 = e^(x - 1). Taking the natural logarithm gives ln((y + 6)/4) = x - 1, so f^-1(x) = 1 + ln((x + 6)/4).

Question 10

An exponential function has the form y = ab^x and passes through the points (0, 5) and (3, 40). Find the values of a and b.

Accepted Answer

Using (0, 5), we get 5 = ab^0, so a = 5. Using (3, 40), we get 40 = 5b^3, so b^3 = 8 and b = 2. The function is y = 5(2)^x.

Question 11

A savings account starts with $1,200 and earns 4.5% annual interest compounded monthly. Write a model for the balance after t years, then find the balance after 6 years.

Accepted Answer

The model is A(t) = 1200(1 + 0.045/12)^(12t). After 6 years, A(6) = 1200(1.00375)^72, which is approximately $1,570.14.

Question 12

A radioactive substance has a half-life of 9 days. If the initial amount is 80 grams, write a model for the amount A after t days and find the amount after 27 days.

Accepted Answer

A half-life model is A(t) = 80(1/2)^(t/9). After 27 days, A(27) = 80(1/2)^3 = 10 grams.

Question 13

Order the functions ln(x), x, and 2^x from slowest growth to fastest growth as x becomes very large. Explain your reasoning.

Accepted Answer

The order from slowest to fastest growth is ln(x), x, and 2^x. Logarithmic functions grow more slowly than linear functions, and exponential functions eventually grow faster than any linear function.

Question 14

For the function y = 2^(-x) + 1, describe whether it represents growth or decay, find the horizontal asymptote, and state the y-intercept.

Accepted Answer

The function represents exponential decay because 2^(-x) is equivalent to (1/2)^x. The horizontal asymptote is y = 1, and the y-intercept is y = 2 because y = 2^0 + 1 = 2 when x = 0.

Question 15

Solve for x: 10^(2x - 1) = 7. Give an exact answer and a decimal approximation.

Accepted Answer

Taking log base 10 of both sides gives 2x - 1 = log(7). Therefore, x = (1 + log(7))/2, which is approximately 0.923.

Precalculus: Exponential and Logarithmic Functions

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