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17Calculus Precalculus - Logarithms Laws

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Logarithm Laws

Here are some laws you need to use when combining logarithms. We use the natural log here because it is the most common logarithm that you will use in calculus but the same rules apply regardless of the base.

1. \( \ln(xy) = \ln(x) + \ln(y) \)

2. \( \ln(x/y) = \ln(x) - \ln(y) \)

3. \( \ln(x^y) = y \ln(x) \)

4. \( e^{\ln(x)} = x \)

5. \( \ln(e) = 1 \)

Let's compare the first couple of laws to exponents.

1. \( e^x e^y = e^{x+y} \)

2. \(\displaystyle{ \frac{e^x}{e^y} = e^{x-y} }\)

Look at the exponents in these two equations and compare them with the corresponding logarithm law. Do you see the similarities? Spend some time comparing them. Write them next to each other on a piece of paper. Thinking about them and turning them over in your mind repeatedly will help you really understand them and know how to use them.

Here is a good video that proves some logarithm properties. It will help you to understand them and how to use them.

PatrickJMT - A Proof of the Logarithm Properties [10min-4secs]

video by PatrickJMT

Examples Using The Logarithm Laws

Before we go on, let's watch this very short video clip explaining a couple of the most common natural log expressions that you will see. They are \(\ln 1\) and \(\ln e \) and then the example \(\ln e^5\).
After those examples, he simplifies \(e^{\ln 7}\), \(e^{3\ln x}\) and \(2e^{4\ln y}\).
At the end of this clip, he simplifies two more expressions that might look complicated at first but they end up to be quite easy to simplify.

The Organic Chemistry Tutor - Logarithms Explained . . . [2min-31secs]

Okay, let's try some practice problems.

Math Word Problems Demystified

Practice

Unless otherwise instructed, simplify, condense or expand these logarithms using the laws above.

\( \log_3 6 - \log_3 2 \)

Problem Statement

Simplify \( \log_3 6 - \log_3 2 \)

Final Answer

\( \log_3 6 - \log_3 2 \) = \( 1 \)

Problem Statement

Simplify \( \log_3 6 - \log_3 2 \)

Solution

The Organic Chemistry Tutor - 4162 video solution

Final Answer

\( \log_3 6 - \log_3 2 \) = \( 1 \)

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\(\log_2 80-\log_2 5\)

Problem Statement

Simplify \(\log_2 80-\log_2 5\)

Solution

Krista King Math - 1542 video solution

video by Krista King Math

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\( \log_3 18 + \log_3 4.5 \)

Problem Statement

Simplify \( \log_3 18 + \log_3 4.5 \) using the rules of logarithms.

Solution

3027 video solution

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\( \log_2 12 + \log_2 24 - \log_2 9 \)

Problem Statement

Simplify \( \log_2 12 + \log_2 24 - \log_2 9 \)

Final Answer

\( \log_2 12 + \log_2 24 - \log_2 9 \) \( = 5 \)

Problem Statement

Simplify \( \log_2 12 + \log_2 24 - \log_2 9 \)

Solution

The Organic Chemistry Tutor - 4163 video solution

Final Answer

\( \log_2 12 + \log_2 24 - \log_2 9 \) \( = 5 \)

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\( \log_7 (1/6) + \log_7 (6/49) \)

Problem Statement

Simplify \( \log_7 (1/6) + \log_7 (6/49) \)

Final Answer

\( \log_7 (1/6) + \log_7 (6/49) \) \( = -2 \)

Problem Statement

Simplify \( \log_7 (1/6) + \log_7 (6/49) \)

Solution

The Organic Chemistry Tutor - 4164 video solution

Final Answer

\( \log_7 (1/6) + \log_7 (6/49) \) \( = -2 \)

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\(\log x - \log y + \log z - \log R\)

Problem Statement

Condense \(\log x - \log y + \log z - \log R\) into a single logarithmic expression.

Final Answer

\(\log x - \log y + \log z - \log R\) = \(\displaystyle{ \log \left( \frac{xz}{yR} \right) }\)

Problem Statement

Condense \(\log x - \log y + \log z - \log R\) into a single logarithmic expression.

Solution

The Organic Chemistry Tutor - 4153 video solution

Final Answer

\(\log x - \log y + \log z - \log R\) = \(\displaystyle{ \log \left( \frac{xz}{yR} \right) }\)

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\(2\log x + 3\log y - 4\log z \)

Problem Statement

Condense \(2\log x + 3\log y - 4\log z \) into a single logarithmic expression.

Final Answer

\(2\log x + 3\log y - 4\log z \) = \(\displaystyle{ \log \left( \frac{x^2y^3}{z^4} \right) }\)

Problem Statement

Condense \(2\log x + 3\log y - 4\log z \) into a single logarithmic expression.

Solution

The Organic Chemistry Tutor - 4157 video solution

Final Answer

\(2\log x + 3\log y - 4\log z \) = \(\displaystyle{ \log \left( \frac{x^2y^3}{z^4} \right) }\)

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\( (1/2)\log x - (1/3)\log y + (1/4)\log z \)

Problem Statement

Condense \( (1/2)\log x - (1/3)\log y + (1/4)\log z \) into a single logarithmic expression.

Final Answer

\( (1/2)\log x - (1/3)\log y + (1/4)\log z \) = \(\displaystyle{ \log \left( \frac{x^{1/2}z^{1/4}}{y^{1/3}} \right) }\)

Problem Statement

Condense \( (1/2)\log x - (1/3)\log y + (1/4)\log z \) into a single logarithmic expression.

Solution

As he mentions in the video, the answer can be written either as \(\displaystyle{ \log \left( \frac{x^{1/2}z^{1/4}}{y^{1/3}} \right) }\) or \(\displaystyle{ \log \left( \frac{\sqrt{x} \sqrt[4]{z}}{\sqrt[3]{y}} \right) }\). In my class, I could prefer the first way, since oftentimes there are more operations to be performed on the expression after this step. However, as usual, check with your instructor to find out what how they want you to write it.

The Organic Chemistry Tutor - 4158 video solution

Final Answer

\( (1/2)\log x - (1/3)\log y + (1/4)\log z \) = \(\displaystyle{ \log \left( \frac{x^{1/2}z^{1/4}}{y^{1/3}} \right) }\)

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\( 3\log x- 5 \log y + (2/3)\log z \)

Problem Statement

Condense \( 3\log x- 5 \log y + (2/3)\log z \) into a single logarithm.

Hint

Remember that when \(\log\) is written without a subscript, the base is assumed to be base 10.

Problem Statement

Condense \( 3\log x- 5 \log y + (2/3)\log z \) into a single logarithm.

Hint

Remember that when \(\log\) is written without a subscript, the base is assumed to be base 10.

Solution

3029 video solution

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\(\displaystyle{ \log \left( \frac{R^2 S^5}{z^6} \right) }\)

Problem Statement

Expand \(\displaystyle{ \log \left( \frac{R^2 S^5}{z^6} \right) }\)

Final Answer

\(\displaystyle{ \log \left( \frac{R^2 S^5}{z^6} \right) }\) = \(\displaystyle{ 2\log R + 5 \log S - 6\log z }\)

Problem Statement

Expand \(\displaystyle{ \log \left( \frac{R^2 S^5}{z^6} \right) }\)

Solution

The Organic Chemistry Tutor - 4159 video solution

Final Answer

\(\displaystyle{ \log \left( \frac{R^2 S^5}{z^6} \right) }\) = \(\displaystyle{ 2\log R + 5 \log S - 6\log z }\)

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\(\displaystyle{ \log \sqrt[3]{ \frac{x^2 y}{z^4} } }\)

Problem Statement

Expand \(\displaystyle{ \log \sqrt[3]{ \frac{x^2 y}{z^4} } }\)

Final Answer

\(\displaystyle{ \log \sqrt[3]{ \frac{x^2 y}{z^4} } }\) = \(\displaystyle{ \frac{1}{3}( 2\log x +\log y - 4\log z ) }\)

Problem Statement

Expand \(\displaystyle{ \log \sqrt[3]{ \frac{x^2 y}{z^4} } }\)

Solution

Since the instructions were to expand the logarithm, I would probably require my students to write the final answer as \( (2/3)\log x + (1/3)\log y - (4/3)\log z \). However, as usual, check with your instructor to see what they require.

The Organic Chemistry Tutor - 4160 video solution

Final Answer

\(\displaystyle{ \log \sqrt[3]{ \frac{x^2 y}{z^4} } }\) = \(\displaystyle{ \frac{1}{3}( 2\log x +\log y - 4\log z ) }\)

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\(\displaystyle{ \log \left( \frac{x^2\sqrt{y}}{\sqrt[3]{z^4}} \right) }\)

Problem Statement

Expand \(\displaystyle{ \log \left( \frac{x^2\sqrt{y}}{\sqrt[3]{z^4}} \right) }\)

Final Answer

\(\displaystyle{ \log \left( \frac{x^2\sqrt{y}}{\sqrt[3]{z^4}} \right) }\) = \(\displaystyle{ 2\log x + \frac{1}{2}\log y - \frac{4}{3}\log z }\)

Problem Statement

Expand \(\displaystyle{ \log \left( \frac{x^2\sqrt{y}}{\sqrt[3]{z^4}} \right) }\)

Solution

The Organic Chemistry Tutor - 4161 video solution

Final Answer

\(\displaystyle{ \log \left( \frac{x^2\sqrt{y}}{\sqrt[3]{z^4}} \right) }\) = \(\displaystyle{ 2\log x + \frac{1}{2}\log y - \frac{4}{3}\log z }\)

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\(\displaystyle{ \log \left[ \frac{\sqrt[3]{ab^2}}{c^4} \right]^5 }\)

Problem Statement

Expand \(\displaystyle{ \log \left[ \frac{\sqrt[3]{ab^2}}{c^4} \right]^5 }\) into sums and/or differences of logarithms.

Solution

3030 video solution

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\( \ln e^{10} - \ln e^4 \)

Problem Statement

Simplify \( \ln e^{10} - \ln e^4 \) using the rules of logarithms.

Solution

3028 video solution

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\( \log_4(256/64) \)

Problem Statement

Simplify \( \log_4(256/64) \)

Solution

3038 video solution

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\( \log_2(128/8) \)

Problem Statement

Simplify \( \log_2(128/8) \)

Solution

3039 video solution

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\(\displaystyle{ \log_2 \left[ \frac{128 \cdot 64}{8 \cdot 16} \right]^5 }\)

Problem Statement

Simplify \(\displaystyle{ \log_2 \left[ \frac{128 \cdot 64}{8 \cdot 16} \right]^5 }\)

Solution

3040 video solution

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\( \log_2(16 \cdot 32) - \log_3(9 \cdot 27) \)

Problem Statement

Simplify \( \log_2(16 \cdot 32) - \log_3(9 \cdot 27) \)

Hint

Be careful to notice that the bases of the two logarithms are different.

Problem Statement

Simplify \( \log_2(16 \cdot 32) - \log_3(9 \cdot 27) \)

Hint

Be careful to notice that the bases of the two logarithms are different.

Solution

3033 video solution

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Really UNDERSTAND Precalculus

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Topics You Need To Understand For This Page

basics of exponentials

basics of logarithms

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Practice Instructions

Unless otherwise instructed, simplify, condense or expand these logarithms using the laws above.

Do NOT follow this link or you will be banned from the site!

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

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