### About Solving Exponential & Logarithmic Equations:

When working with logarithmic and exponential equations, we utilize a variety of techniques to obtain a solution. Solving an exponential equation generally involves taking the log of both sides and then isolating the variable. For the case of logarithmic equations, where two logarithms with the same base are equal, we can set the arguments equal to each other and solve.

Test Objectives

- Demonstrate the ability to solve exponential equations with different bases
- Demonstrate the ability to solve logarithmic equations with logs on each side
- Demonstrate the ability to solve logarithmic equations with a log equal to a number

#1:

Instructions: Solve each equation.

a) $$13^x = 2$$

b) $$9^x = 39$$

c) $$e^p = 72$$

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#2:

Instructions: Solve each equation.

a) $$9 \cdot 20^{10x - 1} + 10 = 103$$

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#3:

Instructions: Solve each equation.

a) $$\log(x) + log(9) = 2$$

b) $$\log{x} - log(2) = log(53)$$

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#4:

Instructions: Solve each equation.

a) $$log_{8}(4x^2 + 7) - log_{8}(2) = 2$$

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#5:

Instructions: Solve each equation.

a) $$ln(7) - ln(7 - 5x) = 3$$

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Written Solutions:

#1:

Solutions:

a) $$x = log_{13}(2) \hspace{.5em} or \hspace{.5em} x = \frac{log(2)}{log(13)}$$

b) $$x = log_{9}(39) \hspace{.5em} or \hspace{.5em} x = \frac{log(39)}{log(9)}$$

c) $$p = ln(72)$$

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#2:

Solutions:

a) $$x = \frac{log_{20}\left(\frac{31}{3}\right) + 1}{10}$$

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#3:

Solutions:

a) $$x = \frac{100}{9}$$

b) $$x = 106$$

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#4:

Solutions:

a) $$x = \pm \frac{11}{2}$$

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#5:

Solutions:

a) $$x = \frac{-7 + 7e^3}{5e^3}$$