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### Section 1-9 : Exponential And Logarithm Equations

For problems 1 – 17 find all the solutions to the given equation. If there is no solution to the equation clearly explain why.

- \(15 = 12 + 5{{\bf{e}}^{10\,w - 7}}\)
- \(4{{\bf{e}}^{2\,x + {x^{\,2}}}} - 7 = 2\)
- \(8 + 3{{\bf{e}}^{4 - 9\,z}} = 1\)
- \(4{t^2} - 3{t^2}{{\bf{e}}^{2 - t}} = 0\)
- \(7x + 16x{{\bf{e}}^{{x^{\,3}} - 5x}} = 0\)
- \(3{{\bf{e}}^{7\,t}} - 12{{\bf{e}}^{8\,t + 5}} = 0\)
- \(2y{{\bf{e}}^{{y^{\,2}}}} - 7y{{\bf{e}}^{1 - 5\,y}} = 0\)
- \(16 + 4\ln \left( {x + 2} \right) = 7\)
- \(\displaystyle 3 - 11\ln \left( {\frac{z}{{3 - z}}} \right) = 1\)
- \(2\log \left( w \right) - \log \left( {3w + 7} \right) = 1\)
- \(\ln \left( {3x + 1} \right) - \ln \left( x \right) = - 2\)
- \(t\log \left( {6t + 1} \right) - 3{t^2}\log \left( {6t + 1} \right) = 0\)
- \(2\log \left( z \right) - \log \left( {{z^2} + 4z + 1} \right) = 0\)
- \(\ln \left( x \right) + \ln \left( {x - 2} \right) = 3\)
- \(11 - {5^{9w - 1}} = 3\)
- \(12 + {20^{7 - 2t}} = 50\)
- \(1 + {3^{{z^2} - 2}} = 5\)

**Compound Interest.** If we put \(P\) dollars into an account that earns interest at a rate of \(r\) (written as a decimal as opposed to the standard percent) for \(t\) years then,

- if interest is compounded \(m\) times per year we will have,
\[A = P{\left( {1 + \frac{r}{m}} \right)^{t\,m}}\]
dollars after \(t\) years.

- if interest is compounded continuously we will have,
\[A = P{{\bf{e}}^{r\,t}}\]
dollars after \(t\) years.

- We have $2,500 to invest and 80 months. How much money will we have if we put the money into an account that has an annual interest rate of 9% and interest is compounded
- quarterly
- monthly
- continuously

- We are starting with $60,000 and we’re going to put it into an account that earns an annual interest rate of 7.5%. How long will it take for the money in the account to reach $100,000 if the interest is compounded
- quarterly
- monthly
- continuously

- Suppose that we put some money in an account that has an annual interest rate of 10.25%. How long will it take to triple our money if the interest is compounded
- twice a year
- 8 times a year
- continuously

**Exponential Growth/Decay.** Many quantities in the world can be modeled (at least for a short time) by the exponential growth/decay equation.

If \(k\) is positive we will get exponential growth and if \(k\) is negative we will get exponential decay.

- A population of bacteria initially has 90,000 present and in 2 weeks there will be 200,000 bacteria present.
- Determine the exponential growth equation for this population.
- How long will it take for the population to grow from its initial population of 90,000 to a population of 150,000?

- We initially have 2 kg grams of some radioactive element and in 7250 years there will be 1.5 kg left.
- Determine the exponential decay equation for this element.
- How long will it take for half of the element to decay?
- How long will it take until there is 250 grams of the element left?

- For a particular radioactive element the value of \(k\) in the exponential decay equation is given by \(k = 0.000825\).
- How long will it take for a quarter of the element to decay?
- How long will it take for half of the element to decay?
- How long will it take 90% of the element to decay?