Q: What is the hydrogen ion concentration of a solution with pH $$7.6$$?

$$[H^+] = 2.51 \times 10$$^{-8}$$ M

Question 1

Expand the logarithmic expression $$\log_{3} \frac{\sqrt{7x}}{2x}$$ using properties of logarithms.

Accepted Answer

$$\frac{1}{2} \log_{3} 7 + \frac{1}{2} \log_{3} x - \log_{3} 2 - \log_{3} x$$

Question 2

Solve for $$x$$: $$3 \log_{3} x + 1 = 7.$$

Accepted Answer

$$x = 216$$

Question 3

Rewrite $$\log_{2} \frac{3}{8} = x as an $$exponential equation.

Accepted Answer

$$2^{x}$$ = \frac{3}{8}$$

Question 4

Afghanistan suffered two major earthquakes in 1998. The first had a magnitude of $$R = 6.1$$ and the second $$R = 6.9. $$Using $$\log \left( \frac{B}{T} \right)$$, how many times more powerful was the second quake? Assume $$B$$ and $$T$$ are equal for both.

Accepted Answer

$$10^{0.8}$$ \approx 6.3$$ times

Question 5

Write the exponential function for a population of $$67,000$$ decreasing at a rate of $$1.67\%$$ per year.

Accepted Answer

$$P(t) = 67000(0.9833$$)^{t}$$

Question 6

Solve for $$x$$: $$\log \sqrt{10} = x.$$

Accepted Answer

$$x = \frac{1}{2} \log 10$$

Question 7

The number $$P of $$students infected with the flu at Olympia High School $$t$$ days after exposure is modeled by $$P(t) = \frac{3000}{1+0.02e$$^{-0.5t}$$}. $$What is the maximum number of students that can be infected?

Accepted Answer

$$3000$$

Question 8

If the pH of sea water is $$7.6$$ and the pH of milk of magnesia is $$10.5$$, how many times greater is the hydrogen ion concentration of sea water than that of milk of magnesia?

Accepted Answer

$$10^{2.9}$$ \approx 794$$ times

Question 9

Given $$1000e^{4x} = 75$, solve for x$.

Accepted Answer

$$x = \frac{1}{4} \ln \left( \frac{75}{1000} \right)$$

Question 10

Expand $$\log(x+2) + \log(x-1) = 4$$ and solve for $$x.$$

Accepted Answer

$$x = 99$$

Question 11

The population of a community (in thousands) is $$P = 224e$$^{0.012t}$$, where t=0$ is the year 2000. Predict the population in the year 2020.

Accepted Answer

$$P \approx 295$$ thousand

Question 12

Given $$4^{x+2}$$ = $$7^{x}$$, solve for $$x.$$

Accepted Answer

$$x = \frac{2 \ln 4}{\ln 7 - \ln 4}$$

Question 13

If $$\frac{50}{4x+2} = 11$$, solve for $$x.$$

Accepted Answer

$$x = 1.045$$

Question 14

Graph the function $$e^{x^2 - 5} - 2 = f(x$$)$$. What is the horizontal asymptote?

Accepted Answer

$$y = -2$$

Question 15

Given $$h(x) = ab$$^{x}$$ + c$$ and the graph passes through $$(0, 4)$$, $$(1, 6)$$, and has a horizontal asymptote at $$y = 2$$, find the values of $$a$$, $$b$$, and $$c.$$

Accepted Answer

$$a = 2$$, $$b = 2$$, $$c = 2$$

Question 16

Is the function $$h(x) = ab$$^{x}$$ + c$$ with $$a \u003e 0$$, $$b \u003e 1$$ exponential growth or decay?

Accepted Answer

Exponential growth, because $$b \u003e 1.$$

Question 17

If the initial population is $$67,000$$ and it increases at a rate of $$1.67\%$$ per year, write the exponential function.

Accepted Answer

$$P(t) = 67000(1.0167$$)^{t}$$

Question 18

What is the hydrogen ion concentration of a solution with pH $$7.6$$?

Accepted Answer

$$[H^+] = 2.51 	imes 10$$^{-8}$$ M

Logarithmic and Exponential Functions: Properties, Applications, and Graphs

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