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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

close solution

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

close solution

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

close solution

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

Problem Statement

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

Solution

1542 video

video by Krista King Math

close solution

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

close solution

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

close solution

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

Problem Statement

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

Solution

3038 video

close solution

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

Problem Statement

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

Solution

3039 video

close solution

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

close solution

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

close solution

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

close solution

Solve \( \ln x = 7 \)

Problem Statement

Solve \( \ln x = 7 \)

Solution

3046 video

close solution

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

close solution

Solve \( 3^x = 5 \)

Problem Statement

Solve \( 3^x = 5 \)

Solution

3035 video

close solution

Solve \( 5^{2x+3} = 8 \)

Problem Statement

Solve \( 5^{2x+3} = 8 \)

Solution

3036 video

close solution

Solve \( 3^{x+2} = 4^{2-x} \)

Problem Statement

Solve \( 3^{x+2} = 4^{2-x} \)

Solution

3037 video

close solution

Solve \( \log_2 16 = x \)

Problem Statement

Solve \( \log_2 16 = x \)

Solution

3041 video

close solution

Solve \( \log_x 81 = 4 \)

Problem Statement

Solve \( \log_x 81 = 4 \)

Solution

3042 video

close solution

Solve \( \log_{32} x = 4/5 \)

Problem Statement

Solve \( \log_{32} x = 4/5 \)

Solution

3043 video

close solution

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

Problem Statement

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

Solution

3044 video

close solution

Solve \( \log x = 24 \)

Problem Statement

Solve \( \log x = 24 \)

Solution

3045 video

close solution

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

Problem Statement

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

Solution

3047 video

close solution

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

Problem Statement

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

Solution

3048 video

close solution

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

Problem Statement

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

Solution

3049 video

close solution

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

Problem Statement

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

Solution

3050 video

close solution

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

Problem Statement

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

Solution

3051 video

close solution

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

Problem Statement

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

Solution

3052 video

close solution

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

close solution

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

close solution

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

close solution

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

close solution

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

close solution

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

close solution

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

close solution

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

close solution

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

video by Khan Academy

close solution

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

close solution

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 \( V(t) = 25e^{kt} \).

Final Answer

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

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 \(x=t\), \(x/t = 1\)

\( \ln[(4/5)] = k \)

Final Answer

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

close solution

Advanced

Solve \( \log x^{\log x} = 49 \)

Problem Statement

Solve \( \log x^{\log x} = 49 \)

Solution

3056 video

close solution

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

Problem Statement

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

Solution

3057 video

close solution

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

Problem Statement

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

Solution

3058 video

close solution

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.

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