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MAT 153: College Algebra and Trigonometry

Subsection 2.9 More Exponents and Logarithms Exercises

  1. Solve the following equations.
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    1. \(\displaystyle 3 \cdot 5^{2x} =1171875\)
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    2. \(\displaystyle 8 - 2^{3x} = -504\)
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    3. \(\displaystyle 3^{4x}-36\cdot3^{2x}+243=0\)
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    4. \(\displaystyle \ln (x+2) + \ln (x-1) = \ln 4\)
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    5. \(\displaystyle \ln (2x+3) + \ln (x-2) = \ln 9\)
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    6. \(\displaystyle 14e^{2x-3}=700\)
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  2. Rewrite each expression using a single natural logarithm.
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    1. \(\displaystyle \ln (x-3) -\ln (x^2-3x)\)
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    2. \(\displaystyle \ln x -\frac{1}{2}\ln (x^2+2x)\)
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    3. \(\displaystyle \ln (x^2z) - 2\ln (z) +\frac{1}{3}\ln(y)\)
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    4. \(\displaystyle 2\ln x - \frac{1}{4}\ln (z) +\frac{2}{3}\ln(y)\)
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  3. The population of the Mexico City metropolitan area was 18.4 million in 2000, and 21.3 million in 2015.
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    1. Write a function equation \(P(t)\) for the number of people living in the Mexico City metropolitan area \(t\) years after 2000.
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    2. Rewrite \(P(t)\) using the natural base, \(e\text{.}\)
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    3. What will the population be in 2020, assuming it continues to grow at the same rate?
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    4. When will the number of people living in the Mexico City metropolitan area reach 30 million? First give an exact answer, and then give a decimal approximation.
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  4. The amount of money accumulated in a bank account where the interest is compounded continually after \(t\) years at interest rate \(r\) with an initial amount \(P\) is given by the function
    \begin{equation*} A(t)=Pe^{rt} \end{equation*}
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    1. If you put $600 in a bank account with interest rate of 3%, compounded continually, write a function equation for \(A(t)\text{.}\)
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    2. How much money will be in the account after 4 years?
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    3. How long will it take the account to reach $1,500?
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    4. Find an equation for the inverse function that gives the time, \(t\) as a function of the amount of money in the account, \(A\text{.}\)
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  5. Refer to the population of Nigeria, for given years as shown in Table 2.9.8.
    Table 2.9.8. Population, \(P(t)\) (in millions of people), of Nigeria \(t\) years after 1990
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    \(t\) 0 5 10 15 20 25
    \(P(t)\) 95.3 108.0 122.4 138.9 158.6 181.2
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    1. Using the population of Nigeria in 1990 and 2015, build an exponential model for \(P(t)\text{.}\)
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    2. Use your exponential model to predict the population of Nigeria in 2010. How does your prediction compare with the actual population at that time?
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    3. Use your exponential model to predict in what year the population of Nigeria will reach 250 million. First give your answer in exact form using a logarithm, and then give the decimal approximation.
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    4. Use \(P(t)\) to predict the population of Nigeria in 2025.
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  6. Let \(f(x)=5\cdot e^{2x}\text{,}\) \(g(x)=\ln (x+1)\) and \(k(x)=x^2\text{.}\)
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    1. Evaluate \(f(g(6))\text{.}\)
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    2. Write an expression for \((g \circ f)x\text{.}\)
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    3. Write an expression for \(g(f(x))k(x)\text{.}\)
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    4. Write an expression for \(f(g(k(x)))\text{.}\)
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    5. For what values of \(x\) is \(f(x) \geq k(x)\text{?}\)
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