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3.8 Concept Review

When the unknown is:  Explain how to solve: Examples you have previously seen (pH, half-life, compound interest, etc.):
The final amount

A
The initial amount

P
The exponent of base e

[latex]e^x[/latex]

 [latex]ln()[/latex] on both sides of the equation
The exponent of base 10

[latex]10^x[/latex]
The exponent of base b

[latex]b^x[/latex]

 Compound interest
The argument of natural log

ln(x)
The argument of common log

log(x)
The argument of log base b

[latex]\log_b(x)[/latex]

 Use base b on both sides
  1. Create a rational function with a vertical asymptote at [latex]x=10[/latex].
  2. Describe what happens to the behavior of a graph when x gets closer to the vertical asymptote.
  3. Explain in your own words what inverse functions are, and give three different examples.
  4. When working with finance problems, what keyword tells you to use the formula [latex]A=Pe^{rt}[/latex]?
  5. For continuous compound interest problems, does P, r, or t have the greatest effect on the final amount?
    (Hint: Make up your own values for P, r, and t, and see which affects the value of A the most.)
  6. Find all errors, and explain how to fix them.
    1. If [latex]\$2000[/latex] is invested at [latex]8.5\%[/latex] compounded monthly for 2 years, then

      [latex]A=2000\left(1+\frac{8.5}{12}\right)^{(12)\;2}[/latex]

    2. If [latex]\$129,000[/latex] is invested at [latex]3.1\%[/latex] compounded weekly for 6 years, then

      [latex]A=129,000\left(1+\frac{0.031}7\right)^{(7)\;\left(6\right)}[/latex]

    3. If [latex]\$800[/latex] is invested at [latex]2.7\%[/latex] interest compounded quarterly for 13 years, then

      [latex]A=800\left(1+0.027/4\right)\wedge4\ast13[/latex]

  7. Explain the difference between the equations below and how to solve each.

    [latex]\ln(6x)=1 [/latex]                                     [latex]6\ln(x)=1[/latex]

  8. Explain the difference between the equations below and how to solve each.

    [latex]10^x-2=5[/latex]                                     [latex]e^x-2=5[/latex]

  9. Which of the following equations require a logarithm to solve? Explain in your own words why or why not.
    1. [latex]50=P\cdot e^{\left(8\right)\left(0.45\right)}[/latex]
    2. [latex]150=5.10^{0.03t}[/latex]
    3. [latex]2300=1300(1+4r)[/latex]
  10. If Topo Chico has a pH of 4.5, find the hydronium ion content.
    1. In January 2020, an earthquake south of San Antonio measured 2.9 on the Richter scale. Find its seismographic reading.
    2. What was the seismographic reading for the October 1989 Loma Prieta earthquake in California if its magnitude was 6.9?
    3. Using your answers from (a) and (b), find how many times stronger the Loma Prieta earthquake was than the one in San Antonio.

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