## 17Calculus Precalculus - Logarithms

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The idea of logarithms is not as complicated as it might seem. Logarithms are just another way to write exponents. It's all about notation. The rules that apply to logarithms can be understood if you keep in mind that you are working with exponents.

If you want a complete lecture on this topic, we recommend this video from one of our favorite instructors.

### Prof Leonard - Exploring the Properties of Logarithms

video by Prof Leonard

Getting Started

You should already be familiar with this expression $$\displaystyle{ 2^3 = 8 }$$. This same expression written as a logarithm is $$\displaystyle{ 3 = \log_2 8 }$$ and is read 'three is the logarithm (think:exponent) base two of eight' or 'three is log (think:power) eight base two'.
You could also say, if I have a base two and I want to get eight, what should the exponent (or power) of the base two be to get eight? The answer is three.

Let's pause for a minute and watch an interesting video talking about a unique way of looking at exponentials and logarithms.

### 3Blue1Brown - Triangle of Power [7min-44secs]

video by 3Blue1Brown

It would be nice if all teachers used the triangle idea when thinking about exponentials and logarithms but things won't change overnight. So we need to study and be able to use the traditional way of looking at logarithms.

In calculus, you will work mostly with logarithms with base $$e$$. These are special logarithms called natural logarithms. The notation is a bit different. Instead of $$\log_e x$$, you will need to write $$\ln(x)$$ or $$\ln ~x$$. It is considered incorrect notation to write $$\log_e x$$.

What Are Logarithms?

Before we go on, here is a great video for you that explains what logarithms are and how they work. It is well worth your time to watch it.

### Dr Chris Tisdell - What are Logarithms? [11min-56secs]

video by Dr Chris Tisdell

Some 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 type logarithm.

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

Okay, let's try some practice problems.

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

What is the domain of the logarithm function $$y = \log_2(5-x) + 3$$?

Problem Statement

What is the domain of the logarithm function $$y = \log_2(5-x) + 3$$?

Solution

### 3031 video

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

Problem Statement

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

Hint

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

Hint: Remember that when $$\log$$ is written without a subscript, the base is assumed to be base 10.

Solution

### 3029 video

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Expand $$\displaystyle{ \log \left[ \frac{\sqrt[3]{ab^2}}{c^4} \right]^5 }$$ into a sum and/or difference of logarithms.

Problem Statement

Expand $$\displaystyle{ \log \left[ \frac{\sqrt[3]{ab^2}}{c^4} \right]^5 }$$ into a sum and/or difference of logarithms.

Solution

### 3030 video

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Simplify $$\log_2 80-\log_2 5$$

Problem Statement

Simplify $$\log_2 80-\log_2 5$$

Solution

### 1542 video

video by Krista King Math

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

Problem Statement

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

Solution

### 3027 video

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

Problem Statement

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

Solution

### 3028 video

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

Problem Statement

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

Solution

### 3038 video

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

Problem Statement

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

Solution

### 3039 video

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

Problem Statement

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

Solution

### 3040 video

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

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

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Graphing Logarithms

Here is a short video discussing how to graph logarithm functions using an example. It is important to have an idea what a logarithm graph looks like. You will need to know this when working with continuity in calculus.

### PatrickJMT - Graphing a Logarithm - Made Easy [5min-22secs]

video by PatrickJMT

Solving Equations Involving Logarithms

When we have a variable in an exponent, we need to move it out of the exponent to determine it's value. To do that, we use the laws listed above. See the practice problems below for examples, then try a few on your own.

Unless otherwise instructed, solve these problems using the natural logarithm giving your answers in exact terms.

Solve $$\log_2(x+1) + \log_2(5x+1) = 6$$

Problem Statement

Solve $$\log_2(x+1) + \log_2(5x+1) = 6$$

Solution

### 3032 video

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Solve $$\ln x = 7$$

Problem Statement

Solve $$\ln x = 7$$

Solution

### 3046 video

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Solve $$\log_3(10x+1) - \log_3(x+1) = 2$$

Problem Statement

Solve $$\log_3(10x+1) - \log_3(x+1) = 2$$

Solution

### 3034 video

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Solve $$3^x = 5$$

Problem Statement

Solve $$3^x = 5$$

Solution

### 3035 video

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Solve $$5^{2x+3} = 8$$

Problem Statement

Solve $$5^{2x+3} = 8$$

Solution

### 3036 video

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Solve $$3^{x+2} = 4^{2-x}$$

Problem Statement

Solve $$3^{x+2} = 4^{2-x}$$

Solution

### 3037 video

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Solve $$\log_2 16 = x$$

Problem Statement

Solve $$\log_2 16 = x$$

Solution

### 3041 video

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Solve $$\log_x 81 = 4$$

Problem Statement

Solve $$\log_x 81 = 4$$

Solution

### 3042 video

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Solve $$\log_{32} x = 4/5$$

Problem Statement

Solve $$\log_{32} x = 4/5$$

Solution

### 3043 video

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Solve $$\log_3(5x+1) = 4$$

Problem Statement

Solve $$\log_3(5x+1) = 4$$

Solution

### 3044 video

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Solve $$\log x = 24$$

Problem Statement

Solve $$\log x = 24$$

Solution

### 3045 video

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Solve $$\log_7 (x^2+3x+9) = 2$$

Problem Statement

Solve $$\log_7 (x^2+3x+9) = 2$$

Solution

### 3047 video

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Solve $$\ln(3x-2) = 5$$

Problem Statement

Solve $$\ln(3x-2) = 5$$

Solution

### 3048 video

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Solve $$4\ln(2x-1) + 3 = 11$$

Problem Statement

Solve $$4\ln(2x-1) + 3 = 11$$

Solution

### 3049 video

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Solve $$\log_3 (5x+2) = \log_3 (7x-8)$$

Problem Statement

Solve $$\log_3 (5x+2) = \log_3 (7x-8)$$

Solution

### 3050 video

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Solve $$\log_2 (x^2+4x) = \log_2 (5)$$

Problem Statement

Solve $$\log_2 (x^2+4x) = \log_2 (5)$$

Solution

### 3051 video

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Solve $$\log_2 x + \log_2 (x+4) = 5$$

Problem Statement

Solve $$\log_2 x + \log_2 (x+4) = 5$$

Solution

### 3052 video

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Solve $$\log_3 (x+1) = 3 - \log_3 (x=7)$$

Problem Statement

Solve $$\log_3 (x+1) = 3 - \log_3 (x=7)$$

Solution

### 3053 video

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Solve $$\log_4 (2x+6) - \log_4 (x-1) = 1$$

Problem Statement

Solve $$\log_4 (2x+6) - \log_4 (x-1) = 1$$

Solution

### 3054 video

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Solve $$\log_2 (x+3) = 4 + \log_2 (x-3)$$

Problem Statement

Solve $$\log_2 (x+3) = 4 + \log_2 (x-3)$$

Solution

### 3055 video

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Unless otherwise instructed, solve $$8^x=15$$ using the natural logarithm giving your answer in exact terms.

Problem Statement

Unless otherwise instructed, solve $$8^x=15$$ using the natural logarithm giving your answer in exact terms.

Solution

### 1698 video

video by PatrickJMT

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Unless otherwise instructed, solve $$7^x-1=4$$ using the natural logarithm giving your answer in exact terms.

Problem Statement

Unless otherwise instructed, solve $$7^x-1=4$$ using the natural logarithm giving your answer in exact terms.

Solution

### 1701 video

video by MIP4U

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Unless otherwise instructed, solve $$3(2^x)-2=13$$ using the natural logarithm giving your answer in exact terms.

Problem Statement

Unless otherwise instructed, solve $$3(2^x)-2=13$$ using the natural logarithm giving your answer in exact terms.

Solution

### 1702 video

video by MIP4U

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Unless otherwise instructed, solve $$(2/3)^x=5^{3-x}$$ using the natural logarithm giving your answer in exact terms.

Problem Statement

Unless otherwise instructed, solve $$(2/3)^x=5^{3-x}$$ using the natural logarithm giving your answer in exact terms.

Solution

### 1703 video

video by MIP4U

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Unless otherwise instructed, solve $$5^{x-3}=3^{2x+1}$$ using the natural logarithm giving your answer in exact terms.

Problem Statement

Unless otherwise instructed, solve $$5^{x-3}=3^{2x+1}$$ using the natural logarithm giving your answer in exact terms.

Solution

### 1704 video

video by MIP4U

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Unless otherwise instructed, solve $$1111=5(2^t)$$ using the natural logarithm giving your answer in exact terms.

Problem Statement

Unless otherwise instructed, solve $$1111=5(2^t)$$ using the natural logarithm giving your answer in exact terms.

Solution

### 1700 video

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Unless otherwise instructed, solve $$\displaystyle{\left( \frac{4}{5} \right)^x = 6^{1-x}}$$ using the natural logarithm giving your answer in exact terms.

Problem Statement

Unless otherwise instructed, solve $$\displaystyle{\left( \frac{4}{5} \right)^x = 6^{1-x}}$$ using the natural logarithm giving your answer in exact terms.

Solution

### 1699 video

video by PatrickJMT

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The exponential function $$V(x) = 25 (4/5)^x$$ is used to model the value of a car over time. Use the properties of logarithms and exponentials to rewrite the model in the form $$V(t) = 25e^{kt}$$.

Problem Statement

The exponential function $$V(x) = 25 (4/5)^x$$ is used to model the value of a car over time. Use the properties of logarithms and exponentials to rewrite the model in the form $$V(t) = 25e^{kt}$$.

$$V(t) = 25e^{t\ln(4/5)}$$

Problem Statement

The exponential function $$V(x) = 25 (4/5)^x$$ is used to model the value of a car over time. Use the properties of logarithms and exponentials to rewrite the model in the form $$V(t) = 25e^{kt}$$.

Solution

Notice that the two equations are the same except for the exponential terms. So we need to determine $$k$$ from $$(4/5)^x = e^{kt}$$. Although not stated in the problem, since the function names are both V, we can assume that $$x=t$$.

 $$(4/5)^x = e^{kt}$$ $$\ln[(4/5)^x] = \ln[e^{kt}]$$ $$x\ln[(4/5)] = kt\ln[e]$$ $$x\ln[(4/5)] = kt$$ $$x\ln[(4/5)] = kt$$ $$(x/t)\ln[(4/5)] = k$$ Since $$x=t$$, $$x/t = 1$$ $$\ln[(4/5)] = k$$

$$V(t) = 25e^{t\ln(4/5)}$$

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Solve $$\log x^{\log x} = 49$$

Problem Statement

Solve $$\log x^{\log x} = 49$$

Solution

### 3056 video

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Solve $$\log x^2 = (\log x)^2$$

Problem Statement

Solve $$\log x^2 = (\log x)^2$$

Solution

### 3057 video

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Solve $$\log (\log x) = 4$$

Problem Statement

Solve $$\log (\log x) = 4$$

Solution

### 3058 video

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Next

Okay, after working those practice problems, you are ready to tackle some application problems involving exponentials and logarithms, starting with exponential growth and decay.

Really UNDERSTAND Precalculus

### Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

$$\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }$$

$$\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }$$

$$\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }$$

$$\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }$$

Set 2 - squared identities

$$\sin^2t + \cos^2t = 1$$

$$1 + \tan^2t = \sec^2t$$

$$1 + \cot^2t = \csc^2t$$

Set 3 - double-angle formulas

$$\sin(2t) = 2\sin(t)\cos(t)$$

$$\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }$$

Set 4 - half-angle formulas

$$\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }$$

$$\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }$$

Trig Derivatives

 $$\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }$$ $$\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }$$ $$\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }$$ $$\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }$$ $$\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }$$ $$\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }$$

Inverse Trig Derivatives

 $$\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }$$ $$\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }$$

Trig Integrals

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

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