## 17Calculus Precalculus - Logarithms Laws

##### 17Calculus

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.

Practice

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

$$\log_3 6 - \log_3 2$$

Problem Statement

Simplify $$\log_3 6 - \log_3 2$$

$$\log_3 6 - \log_3 2$$ = $$1$$

Problem Statement

Simplify $$\log_3 6 - \log_3 2$$

Solution

### The Organic Chemistry Tutor - 4162 video solution

$$\log_3 6 - \log_3 2$$ = $$1$$

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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 $$\log_2 12 + \log_2 24 - \log_2 9$$

$$\log_2 12 + \log_2 24 - \log_2 9$$ $$= 5$$

Problem Statement

Simplify $$\log_2 12 + \log_2 24 - \log_2 9$$

Solution

### The Organic Chemistry Tutor - 4163 video solution

$$\log_2 12 + \log_2 24 - \log_2 9$$ $$= 5$$

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

Problem Statement

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

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

Problem Statement

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

Solution

### The Organic Chemistry Tutor - 4164 video solution

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

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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.

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

Problem Statement

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

Solution

### The Organic Chemistry Tutor - 4153 video solution

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

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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.

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

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

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

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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.

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

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

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

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

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

Problem Statement

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

Solution

### The Organic Chemistry Tutor - 4159 video solution

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

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

Problem Statement

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

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

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

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

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

Problem Statement

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

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

Problem Statement

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

Solution

### The Organic Chemistry Tutor - 4161 video solution

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

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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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