Logarithmic Functions Their Graphs And Applications¶
Rewriting exponentials into logarithms and logarithms into exponentials using common log, natural log, and logarithms of other bases¶
Source: I made these up.
Explanation:
This artifact demonstrates rewriting exponentials into logarithms and logarithms into exponentials using common log, natural log, and logarithms of other bases.
- Exponential \(\rightarrow\) Common Log
- Commong log \(\rightarrow\) Exponential
- Exponential \(\rightarrow\) Natural Log
- Natural log \(\rightarrow\) exponential
Artifact:
- \(4^x = 6 \rightarrow log_4 6 = x\)
- \(log_2 2 = x \rightarrow 2^x = 2\)
- \(e^x = e \rightarrow ln(e)\)
- \(ln(8) \rightarrow e^x = 8\)
Properties of logarithms¶
Source: Notes
Explanation:
This artifact demonstrates properties of logarithms.
- This problem demonstrates the Product Rule.
\(log_b {rs} = log_b r + log_b s\)
- This problem demonstrates the Quotient Rule.
\(log_b {r \over s} = log_b r - log_b s\)
- This problem demonstrates the Power Rule.
\(log_b {r^c} = log_b r * c\)
Artifact:
- \(log(x+6) + log(x-2) = 2\)
- \(log(x+6) - log(x-2) = 2\)
- \(\text{Solve for }log_4 117\)
Graphs of logarithms¶
Source: Made it up.
Explanation:
This artifact demonstrates graphs of logarithms.
I started with the base function \(y = log(x)\) and manipulated it into \(log(x-3)+1\).
The formula \(y = log(x)\) is the same as \(10^y = x\), which is easier to evaluate (for y).
Artifact:
Graph \(log(x-3)+1\)
x | y |
---|---|
0.01 | -2 |
0.1 | -1 |
1 | 0 |
10 | 1 |
x+3 | y+1 |
---|---|
3.01 | -1 |
3.1 | 0 |
4 | 1 |
13 | 2 |
Applications of logarithms¶
Source: #53 from Section 3.4
Explanation:
This artifact demonstrates applications of logarithms.
In the first step I demonstrate that I know how to re-write common logs into exponential form.
After that, I can plug in the given x value (40 ft) and the equation becomes linear and easy to solve.
Awareness and Appreciation:
In this artifact I demonstrate that I am aware that I am error-prone even if the problem is easy, and that I can appreciate double-checking my answers.
Artifact:
The relationship between intensity I of light (in lumens) at a depth of x feet in Lake Superior is given by \(log({I \over 12}) = -0.00235x\)
What is the intensity at a depth of 40 ft?