3.5E: Logarithmic Properties (Exercises)
Part 1: Working with logarithmic functions
1. For each function find the domain of the logarithmic function. Then match each function with a graph.
a. \(f(x)= \log _{3} (x) \) b. \(g(x)= \log _{7} (2x+1) \) c. \(h(x)= ln(7-3x) \) d. \(k(x)= \log _{2} (4-x) \)
2. Simplify the logarithmic expressions of a single variable by writing them as a single logarithm.
a. \( \log _{2} \left(5x\right)+\log _{2} \left(8x^{3}\right)-\log _{2} \left(2x^{2}\right)\)
b. \(3ln \left(x^3\right)+2ln \left(x^{4}\right)\)
c. \( 5\log _{3} \left(5x^5\right)-4\log _{3} \left(2x^{2}\right)\)
d. \(2\ln \left(5x^4\right)+4\ln \left(3x^{5}\right)-3\ln \left(7x^{2}\right)\)
3. Simplify the logarithmic expressions of multiple variables by writing them as a sum or difference of simpler logarithms.
a. \( \log _{2} \left (x^{3}y^{4}\right)\)
b. \( \log _{5} \left (\dfrac{z^{7}}{y^{2}}\right)\)
c. \(ln \left(\sqrt{a}\sqrt[3]{b}\right)\)
d. \(log \left(\dfrac{xy}{z^{2}}\right)\)
4. Write a sentence explaining the mistake that was made in the simplification below. Then correctly simplify if possible.
\(4ln \left(3\right)-ln \left(9\right)\)
To simplify:
Apply the log quotient property \(\dfrac {4ln \left(3\right)}{ln \left(9\right)}\)
Write 9 as a power of 3 \(\dfrac {4ln \left(3\right)}{ln \left(3^{2}\right)}\)
Apply the power property of logs \(\dfrac {4ln \left(3\right)}{2ln \left(3\right)}\)
Simplify \(\dfrac {4}{2} = 2\)
5. Write a sentence explaining the mistake that was made in the simplification below. Then correctly simplify if possible.
\(log \left(5\right)+2log \left(4\right)-3log \left(2\right)\)
To simplify:
Apply the power property of logs \(log \left(5\right)+log \left(4^{2}\right)-log \left(2^{3}\right)\)
Calculate \(log \left(5\right)+log \left(16\right)-log \left(8\right)\)
Apply the sum & difference properties of logs \(log \left(5+16-8\right)\)
Simplify \(log \left(13\right)\)
6. Write a sentence explaining the mistake that was made in the simplification below. Then correctly simplify if possible.
\(2\log_{2} \left(x^{4}\right)+3\log_{2}\left(4x^{2} \right)\)
To simplify:
Apply the power property of logs \(\log_{2} \left((x^{4})^{2}\right)+\log_{2}\left(4(x^{2})^{3} \right)\)
Apply the power property of exponents \(\log_{2} \left(x^{8}\right)+\log_{2}\left(4x^{6}\right)\)
Apply the sum property of logs \(\log_{2} \left(x^{8}4x^{6}\right)\)
Simplify \(\log_{2} \left(4x^{14}\right)\)
7. Solve the logarithmic equations.
a. \(\log_{2} (x) + \log_{2} (3x)=5\)
b. \(ln \left(x^{3}\right)-ln \left(x\right)=1\)
c. \(\log_{5} (x-4) + \log_{5} (x)=1\)
d. \(3\log_{4} (x) + 5\log_{4} (3x)=7\)
e. \(log \left(x+4\right)-log \left(x-2\right)=1\)
8. Write a sentence explaining the mistake that was made in the solving of the equation below. Then correctly solve the equation.
\(5e^{x}=7\)
To solve:
Take the ln of both sides \(ln \left(5e^{x}\right)=ln \left(7\right)\)
Apply the inverse property of logs and exp. \(5x =ln \left(7\right)\)
Apply the sum property of logs \(x = \dfrac{ln \left(7\right)}{5} \approx 0.3892\)
9. Write a sentence explaining the mistake that was made in the solving of the equation below. Then correctly solve the equation.
\(ln \left(x\right)+ln \left(5x\right)=1\)
To solve:
Apply the sum property of logs \(ln \left(x+5x\right)=1\)
Simplify \(ln \left(6x\right)=1\)
Convert to exponential form \(e^{1}=6x\)
Isolate x \(x=\dfrac{e}{6} \approx 0.4530\)
Part 2: Applications
10. The 1906 San Francisco earthquake had a magnitude of 7.9 on the MMS scale. Later there was an earthquake with magnitude 4.7 that caused only minor damage. How many times more intense was the San Francisco earthquake than the second one?
11. The 1906 San Francisco earthquake had a magnitude of 7.9 on the MMS scale. Later there was an earthquake with magnitude 6.5 that caused less damage. How many times more intense was the San Francisco earthquake than the second one?
12. One earthquake has magnitude 3.9 on the MMS scale. If a second earthquake has 750 times as much energy as the first, find the magnitude of the second quake.
13. One earthquake has magnitude 4.8 on the MMS scale. If a second earthquake has 1200 times as much energy as the first, find the magnitude of the second quake.
14. Determine the hydrogen ion concentration of a liquid with pH of 8.
15. Determine the hydrogen ion concentration of an acidic liquid with pH of 4.
16. When will an investment of $10,000 growing by 5% per year be larger in value that an investment of $12,000 that is growing by only 3% per year?
17. A city with a population of 210,000 people is growing by 2.5% per year. Another city with a population of 135,000 is growing by 8% per year. When will the cities have the same population if these trends continue?
- Answer
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Part 1:
1. a. \((0,\infty)\) Graph C c. \((-\infty,\dfrac{7}{3})\) Graph B
2 a. \( \log _{2} \left(20x^2)\right)\) c. \(ln \left(\dfrac{3125x^{17}}{16}\right)\)
3. a. \( 3\log _{2} \left(x\right)+2\log _{2} \left(y\right)\) c. \(\dfrac{1}{2} ln\left(a\right)+\dfrac{1}{3} \left(b\right)\)
5. The sum and difference properties of logs were applied incorrectly in the third step. To add log terms of the same base, we multiply the inputs. To subtract log terms of the same base, we divide the inputs. The last two steps should read \[log \left(\dfrac{5 \cdot 16}{8}\right) = log \left(10\right) = 1 \nonumber\]
7. a. \(x = \pm 1\) c. \(x = -1, x = 5\) e. \(\dfrac{24}{9}\)
9. The sum property of logs was applied incorrectly in the first step. To add log terms of the same base, we multiply the inputs. The first step should read \[ln \left(x \cdot 5x\right) = ln \left(6x^{2}\right) \nonumber\]
Part 2:
11. The San Francisco earthquake's earth movement is 125.9 times as large as than the second earthquake.
13. The second quake has a magnitude of 6.85.
15. \(\left[H^{+} \right]=10^{-4} =0.0001\text{ moles per liter} \)
17. \(x \approx 8.441\)