Q: A population of bacteria doubles every 4 hours. If the initial population is 500, what will the population be after 12 hours?

$$P = 500 \times 2$$^{12/4}$$ = 4000$$

Question 1

Solve the logarithmic equation: $$7 \log_9$(x) - 4 \log_{20}(x) = 7. $$What is the value of $$x$$?

Accepted Answer

$$x = 20$$^{7/4}$$ \cdot 9$$^{-1}$$

Question 2

Which of the following is the correct solution to the equation $$2 \log_5$(m) - \log_5$(12) = 0$$?

Accepted Answer

$$m = 6$$

Question 3

A scientist is analyzing the pH of a solution, which is defined as $$	ext{pH} = -\log_{10}([H^+]). If $$the hydrogen ion concentration $$[H^+] is$$ $$1 	imes 10$$^{-7}$$, what is the pH?

Accepted Answer

$$7$$

Question 4

Solve for $$a in $$the equation $$\log_3$(4) + \log_3$(a) + \log_3$(5) = \log_3$(56). $$Which of the following is correct?

Accepted Answer

$$a = 56/20$$

Question 5

Given $$4 \log_5$(m + 9) = 3$$, what is the value of $$m$$?

Accepted Answer

$$m = 5$$^{3/4}$$ - 9$$

Question 6

Solve the exponential equation: $$2^{25 - v} = 625$. What is the value of v$?

Accepted Answer

$$v = 22$$

Question 7

A radioactive substance decays according to $$N(t) = N_0$ e$$^{-kt}$$. If N_0$ = 100$$ and after $$t = 3$$ hours, $$N(3) = 50$$, what is the decay constant $$k$$?

Accepted Answer

$$k = \frac{\ln(2)}{3}$$

Question 8

Solve for $$x in $$the equation $$4^{2x - 1} = 8$$ \cdot 16^x$$. What is the value of x$?

Accepted Answer

$$x = 1$$

Question 9

If $$8^k = 78$$, what is the value of $$k$$? Round your answer to the nearest ten-thousandth.

Accepted Answer

$$k = \frac{\ln(78)}{\ln(8)} \approx 1.9727$$

Question 10

Solve for $$m in $$the equation $$6^9$$ = $$78^{m - 2}. $$What is the value of $$m$$? Round to the nearest ten-thousandth.

Accepted Answer

$$m = 2 + \frac{9 \ln(6)}{\ln(78)} \approx 4.0547$$

Question 11

Solve for $$a in $$the equation $$3^{15 + 7a} = 67$. What is the value of a$? Round to the nearest ten-thousandth.

Accepted Answer

$$a = \frac{\ln(67) - 15 \ln(3)}{7 \ln(3)} \approx -1.7712$$

Question 12

Solve for $$x in $$the equation $$3^8 + 14$$ \cdot 9^x = 77$$. What is the value of x$? Round to the nearest ten-thousandth.

Accepted Answer

$$x = \frac{77 - 6561}{14 \cdot 9}$$

Question 13

Solve for $$n in $$the equation $$8^3 - 21$$ \cdot $$51^{n - 1} = 0$. What is the value of n$?

Accepted Answer

$$n = 1 + \frac{\ln(512/21)}{\ln(51)}$$

Question 14

Solve for $$r in $$the equation $$10^3 - 2$$ \cdot $$18^{1 - r} = 73$. What is the value of r$? Round to the nearest ten-thousandth.

Accepted Answer

$$r = 1 - \frac{\ln((1000 - 73)/2)}{\ln(18)} \approx -0.0477$$

Question 15

Which property allows you to combine $$\log_b(M) + \log_b(N)$$ into a single logarithm?

Accepted Answer

$$\log_b(MN)$$

Question 16

A population of bacteria doubles every 4 hours. If the initial population is 500, what will the population be after 12 hours?

Accepted Answer

$$P = 500 	imes 2$$^{12/4}$$ = 4000$$

Question 17

Solve for $$k in $$the equation $$6 \log_7$(k - 1) = 3. $$What is the value of $$k$$?

Accepted Answer

$$k = 1 + 7$$^{0.5}$$

Question 18

If $$\log_8$(n) + \log_4$(2) = 0$$, what is the value of $$n$$?

Accepted Answer

$$n = 1/2$$

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

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