### About Logarithmic Functions:

We can say that: y = log_{a}(x) is the same as: x = a^{y}. So for all intents and purposes, a logarithm is an exponent. When we see log_{a} (x), we are asking for the exponent to which the base (a) must be raised to obtain (x). As an example, suppose we see: log_{2} (8). We are asking what exponent must the base (2) be raised to, in order to obtain 8. The answer is 3, since 2^{3} = 8 : log_{2} (8) = 3. We need to know how to convert between exponential and logarithmic form. The process is fairly simple, we just need to understand what is being isolated in each scenario. In exponential form: 3^{2} = 9, here 9, the power is isolated. In logarithmic form, we have: log_{3} (9) = 2, here 2, the exponent is isolated. We solve simple logarithmic equations by converting into exponential form and solving the resulting equation.

Test Objectives

- Demonstrate an understanding of logarithms
- Demonstrate the ability to change between logarithmic form and exponential form
- Demonstrate the ability to solve simple logarithmic equations

#1:

Instructions: evaluate each.

$$a)\hspace{.2em}log_{343}\left(\frac{1}{7}\right)$$

$$b)\hspace{.2em}log_{3}\left(\frac{1}{81}\right)$$

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#2:

Instructions: evaluate each.

$$a)\hspace{.2em}log_{5}\left(\frac{1}{125}\right)$$

$$b)\hspace{.2em}log_{3}(81)$$

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#3:

Instructions: solve each equation.

$$a)\hspace{.2em}log_{x}(6)=\frac{1}{3}$$

$$b)\hspace{.2em}log_{4}(x)=2$$

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#4:

Instructions: solve each equation.

$$a)\hspace{.2em}log_{3}(243)=x$$

$$b)\hspace{.2em}log_{6}(-36)=x$$

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#5:

Instructions: solve each equation.

$$a)\hspace{.2em}5 - 9log_{5}(-4x - 1)=14$$

$$b)\hspace{.2em}4log_{4}(-3x - 6) - 3=9$$

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Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}-\frac{1}{3}$$

$$b)\hspace{.2em}-4$$

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#2:

Solutions:

$$a)\hspace{.2em}-3$$

$$b)\hspace{.2em}4$$

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#3:

Solutions:

$$a)\hspace{.2em}x=216$$

$$b)\hspace{.2em}x=16$$

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#4:

Solutions:

$$a)\hspace{.2em}x=5$$

$$b)\hspace{.2em}No \hspace{.2em}Solution$$

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#5:

Solutions:

$$a)\hspace{.2em}x=-\frac{3}{10}$$

$$b)\hspace{.2em}x=-\frac{70}{3}$$