### About Common & Natural Logarithms:

We can use the change of base rule to change the base of a logarithm into one that is more convenient to work with. We can use this method to obtain a common logarithm or natural logarithm. These two types of logarithms appear on most calculators and can be used to obtain a decimal approximation.

Test Objectives

- Demonstrate the ability to approximate the value of a common logarithm
- Demonstrate the ability to approximate the value of a natural logarithm
- Demonstrate the ability to use the change of base rule to generate a common or natural logarithm

#1:

Instructions: Use the change of base rule and a calculator to approximate each to 3 decimal places.

a) $$\log_{7}(1.17)$$

b) $$\log_{3}(3.7)$$

c) $$\log_{5}(25)$$

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#2:

Instructions: Use the change of base rule and a calculator to approximate each to 3 decimal places.

a) $$\log_{6}(70)$$

b) $$\log_{6}(21)$$

c) $$\log_{12}(\sqrt[4]{12})$$

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#3:

Instructions: Use the change of base rule and a calculator to approximate each to 3 decimal places.

a) $$\log_{3}(81)$$

b) $$\log_{7}(27)$$

c) $$\log_{2}(26)$$

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#4:

Instructions: Use the change of base rule and a calculator to approximate each to 3 decimal places.

a) $$\log_{5}(33)$$

b) $$\log_{3}(3)$$

c) $$\log_{5}(2)$$

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#5:

Instructions: Use the change of base rule and a calculator to approximate each to 3 decimal places.

a) $$\log_{5}(-22)$$

b) $$\log_{2}(29)$$

c) $$\log_{6}(\sqrt[8]{6})$$

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Written Solutions:

#1:

Solutions:

a) $$log_{7}(1.17) = \frac{log(1.17)}{log(7)} \approx .081$$

b) $$log_{3}(3.7) = \frac{log(3.7)}{log(3)} \approx 1.191$$

c) $$log_{5}(25)= \frac{log(25)}{log(5)} = 2$$

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#2:

Solutions:

a) $$log_{6}(70) = \frac{log(70)}{log(6)} \approx 2.371$$

b) $$log_{6}(21) = \frac{log(21)}{log(6)} \approx 1.699$$

c) $$log_{12}(\sqrt[4]{12})= log_{12}(12^{\frac{1}{4}}) = \frac{1}{4}$$

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#3:

Solutions:

a) $$log_{3}(81) = \frac{log(81)}{log(3)} = 4$$

b) $$log_{7}(27) = \frac{log(27)}{log(7)} \approx 1.694$$

c) $$log_{2}(26) = \frac{log(26)}{log(2)} \approx 4.7$$

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#4:

Solutions:

a) $$log_{5}(33) = \frac{ln(33)}{ln(5)} \approx 2.173$$

b) $$log_{3}(3) = \frac{log(3)}{log(3)} = 1$$

c) $$log_{5}(2) = \frac{ln(2)}{ln(5)} \approx .431$$

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#5:

Solutions:

a) $$undefined$$

b) $$log_{2}(29) = \frac{ln(29)}{ln(2)} \approx 4.858$$

c) $$log_{6}(\sqrt[8]{6}) = log_{6}(6^\frac{1}{8}) = \frac{1}{8}$$