Question 1

A biologist models the population of bacteria in a petri dish as $$P(t) = 500 \cdot 2^t$$, where $$t is $$the number of hours. After how many hours will the population reach 4000?

Accepted Answer

$$t = 4$$

Question 2

Solve the exponential equation $$4^{x-2} = 64$ for x$.

Accepted Answer

$$x = 4$$

Question 3

A radioactive substance decays according to $$N(t) = N_0$ e$$^{-0.05t}$$. If the initial amount is N_0$ = 1000$$, after how many years will only $$250$$ remain?

Accepted Answer

$$t = 27.73$$

Question 4

Solve for $$x$$: $$3^{2x} = 27$

Accepted Answer

$$x = 1.5$$

Question 5

A medical researcher uses the formula $$C = 10e$$^{0.2t}$$ to model the concentration of a drug in the bloodstream, where t$ is in hours. After how many hours will the concentration reach 40$ mg/L?

Accepted Answer

$$t = 6.93$$

Question 6

Solve the logarithmic equation $$\log_2$ 3x = -1$$ for $$x.$$

Accepted Answer

$$x = \frac{1}{6}$$

Question 7

Solve for $$x$$: $$\log_4$ (x+1) - \log_4$ (x-1) = 2$$

Accepted Answer

$$x = 5$$

Question 8

A chemical reaction rate is modeled by $$R = 5e$$^{0.1t}$$. If the rate reaches 20$, how long has the reaction been running?

Accepted Answer

$$t = 13.86$$

Question 9

Solve for $$x$$: $$\log_3$ (x+14) - \log_3$ x = \log_3$ 6$$

Accepted Answer

$$x = 2$$

Question 10

Solve for $$x$$: $$\ln(x+4) = 2$$

Accepted Answer

$$x = e^2$ - 4$$

Question 11

Solve for $$x$$: $$3\log_{10} x + \log_{10} 4 = 14$$

Accepted Answer

$$x = 10$$^{\frac{14 - \log_{10}$$ 4}{3}}$$

Question 12

Solve for $$x$$: $$\log_5$ (x+4) = 1$$

Accepted Answer

$$x = 4$$

Question 13

A savings account grows according to $$A = 1000e$$^{0.03t}$$. After how many years will the account reach 2000$?

Accepted Answer

$$t = \frac{\ln 2}{0.03}$$

Question 14

Solve for $$x$$: $$\log_2$ (x+1) + \log_2$ (x-1) = 4$$

Accepted Answer

$$x = 5$$

Question 15

Solve for $$x$$: $$e^{2x} = 200$

Accepted Answer

$$x = \frac{1}{2} \ln 200$$

Question 16

Solve for $$x$$: $$\log_3$ (x+6) + \log_3$ (x) = \log_3$ 14$$

Accepted Answer

$$x = 2$$

Question 17

A population doubles every 5 years. If the initial population is $$1000$$, how long will it take to reach $$8000$$?

Accepted Answer

$$t = 15$$

Question 18

Solve for $$x$$: $$\log_5$ (x+4) + 1 = 4$$

Accepted Answer

$$x = 121$$

Question 19

Solve for $$x$$: $$\log_2$ (x+1) = 3$$

Accepted Answer

$$x = 7$$

Question 20

Solve for $$x$$: $$\log_4$ (x+1) + \log_4$ (x-1) = 3$$

Accepted Answer

$$x = 5$$

Step-by-Step Guidance for Solving Exponential and Logarithmic Equations

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