Question 1

Solve for $$x in $$the equation:

$$4^{x-7}$$ \cdot $$2^{x+12} = 0$

Accepted Answer

No solution; the equation has no real solution.

Question 2

Which of the following is the correct equation to express: 'At time $$t$$, the patient has 8 mg of atorvastatin in their body,' given $$a(t) = 20 \cdot \left(\frac{1}{2}\$$right)^{t/14}$$?

Accepted Answer

$$8 = 20 \cdot \left(\frac{1}{2}\$$right)^{t/14}$$

Question 3

Given $$a(t) = 20 \cdot \left(\frac{1}{2}\$$right)^{t/14}$$, solve for t$ when a(t) = 8$. What is the value of t$?

Accepted Answer

$$t = 14 \cdot \frac{\ln\left(\frac{8}{20}\right)}{\ln\left(\frac{1}{2}\right)}$$

Question 4

A turkey is placed in an oven at $$350^\circF. $$Its temperature after $$t$$ hours is given by $$T(t) = 350 - 290e$$^{-0.16t}$$. What equation would you set up to find when the turkey reaches 165$$^\circ$$F?

Accepted Answer

$$165 = 350 - 290e$$^{-0.16t}$$

Question 5

Rearrange the equation $$165 = 350 - 290e$$^{-0.16t}$$ so that e$$^{-0.16t}$$ is isolated on one side.

Accepted Answer

$$e^{-0.16t}$$ = \frac{350 - 165}{290}$$

Question 6

Solve for $$t in$$ $$e^{-0.16t}$$ = \frac{185}{290}$$. What is the correct expression for t$?

Accepted Answer

$$t = -\frac{1}{0.16} \ln\left(\frac{185}{290}\right)$$

Question 7

What happens if you try to solve $$350 - 290e$$^{-0.16t}$$ = 375$$ for $$t$$? What does this mean physically?

Accepted Answer

You get a negative value inside the logarithm, which is not possible; the turkey cannot reach $$375^\circF.$$

Question 8

Solve for $$x in $$the equation:

$$\ln(x+3) - \ln(x) = \ln(74)$$

Accepted Answer

Question 9

Solve for $$x in $$the equation:

$$\ln(x+3) + \ln(x) = \ln(70)$$

Accepted Answer

$$x = 7$$

Question 10

An investment starts at $$10,000 and increases by 4% per year. How long will it take for the investment to be worth at least 25$$,000? Which equation correctly models this scenario?

Accepted Answer

$$25,000 = 10,000 \cdot (1.04)^t$$

Question 11

Given $$25,000 = 10,000 \cdot (1.04)^t$$, solve for $$t.$$

Accepted Answer

$$t = \frac{\ln(2.5)}{\ln(1.04)}$$

Question 12

Solve for $$x in $$the equation:

$$2\log_{10}(8x+4) + 6 = 10$$

Accepted Answer

$$x = 3$$

Question 13

Solve for $$x in $$the equation:

$$\log_{10}(x) = \log_{10}(x) + 1$$

Accepted Answer

No solution; the equation is inconsistent.

Question 14

Which of the following best describes the process for solving $$\ln(x+3) - \ln(x) = \ln(74)$$?

Accepted Answer

Combine the logarithms using the quotient rule, then exponentiate both sides.

Question 15

A patient’s medication decays according to $$a(t) = 20 \cdot \left(\frac{1}{2}\$$right)^{t/14}$$. If the patient takes a dose every 14 hours, what is the amount of medication just before the next dose?

Accepted Answer

$$a(14) = 10 mg$$

Question 16

If you graph $$T(t) = 350 - 290e$$^{-0.16t}$$, what is the horizontal asymptote and what does it represent in the context of the turkey problem?

Accepted Answer

The horizontal asymptote is $$T = 350$$, representing the oven temperature.

Question 17

Which property of logarithms allows you to combine $$\ln(x+3) + \ln(x)$$ into a single logarithm?

Accepted Answer

Product property: $$\ln(a) + \ln(b) = \ln(ab)$$

Step-by-Step Guidance for Exponential and Logarithmic Equations (Precalculus)

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