Unit 4 · Topic 2.9
Logarithmic Expressions
Interpret logarithms as exponents and evaluate expressions using equivalent exponential statements. Develop the idea through symbolic, numerical, graphical, and contextual representations.
Learning Goals
- Interpret logarithms as exponents.
- Convert between exponential and logarithmic forms.
- Evaluate exact values and enforce restrictions.
1. Essential Structure
Interpret logarithms as exponents and evaluate expressions using equivalent exponential statements.
Read the formula together with its domain, units, starting input, and the interval length over which change is measured.
2. Core Ideas
- \(\log_b c=a\) means \(b^a=c\).
- Real logarithms require \(b>0\), \(b\ne1\), and \(c>0\).
- \(\log_b1=0\), \(\log_bb=1\), and \(\log_b(b^x)=x\).
- When no base is written, \(\log x\) denotes the common logarithm with base 10.
3. Graph and Representation
Use graphical evidence together with an algebraic or numerical reason. A graphing window alone does not establish domain, end behavior, or model validity.
4. Original Worked Example
Evaluate \(\log_2 8\).
Check the result in the original representation and explain its meaning, including units when the quantities are contextual.
5. AP Reasoning Workflow
- Identify base, argument, and exponent.
- Rewrite as an exponential equation.
- Express the argument as a power when possible.
- Solve for the exponent.
- Check restrictions.
A strong AP response shows the mathematical evidence first and then states a precise conclusion.
6. Extended Example and Application
To evaluate \(\log_{1/3}27\), solve \((1/3)^x=3^3\). Since \(3^{-x}=3^3\), the value is \(-3\).
7. Technology and Validation
Use a graphing utility or calculator to estimate nonstandard logarithmic values, but preserve exact values when powers are recognizable.
When technology is used, record the model or result, preserve sufficient precision, and explain why the output answers the question.
8. Read the Three Parts of a Logarithm
A logarithm answers an exponent question:
| Symbol | Role | Condition for real logarithms |
|---|---|---|
| \(b\) | Base | \(b>0\) and \(b\ne1\) |
| \(c\) | Argument | \(c>0\) |
| \(a\) | Logarithmic value, or exponent | Any real number is possible |
For \(\log_4(64)=3\), the base is 4, the argument is 64, and the value 3 says that three factors of 4 produce 64.
Base 1 is excluded because \(1^x=1\) for every exponent, so it cannot identify a unique exponent for general positive arguments. Negative bases are excluded in the real logarithmic function because arbitrary real exponents would not consistently produce real values.
9. Translate between Logarithmic and Exponential Forms
The two forms describe the same relationship from different directions. Keep the base unchanged.
| Logarithmic form | Equivalent exponential form | Reading |
|---|---|---|
| \(\log_2(32)=5\) | \(2^5=32\) | 2 raised to 5 gives 32 |
| \(\log_7(1)=0\) | \(7^0=1\) | 7 raised to 0 gives 1 |
| \(\log_3(1/27)=-3\) | \(3^{-3}=1/27\) | A negative exponent gives a reciprocal |
| \(\log_{16}(4)=1/2\) | \(16^{1/2}=4\) | A fractional exponent gives a root |
When converting \(5^x=19\), the exponent becomes the logarithmic value: \(x=\log_5(19)\). The argument is 19, not \(x\), because 19 is the output of the exponential expression.
10. Evaluate Exact Logarithmic Values
To evaluate without technology, rewrite the argument as a recognizable power of the base.
| Expression | Exponent question | Value |
|---|---|---|
| \(\log_6(216)\) | \(6^{?}=216=6^3\) | 3 |
| \(\log_9(1/81)\) | \(9^{?}=9^{-2}\) | -2 |
| \(\log_{25}(5)\) | \((5^2)^{?}=5\) | \(1/2\) |
| \(\log_{1/2}(8)\) | \((2^{-1})^{?}=2^3\) | -3 |
Two anchor values work for every valid base:
The first follows from \(b^0=1\); the second follows from \(b^1=b\).
11. Understand Bases between Zero and One
A logarithm may have a base between 0 and 1. Such an exponential base decreases as its exponent increases, which reverses several familiar sign patterns.
| Argument \(c\) | If \(b>1\) | If \(0<b<1\) |
|---|---|---|
| \(c>1\) | \(\log_b c>0\) | \(\log_b c<0\) |
| \(c=1\) | \(\log_b c=0\) | \(\log_b c=0\) |
| \(0<c<1\) | \(\log_b c<0\) | \(\log_b c>0\) |
For example, \((1/3)^{-2}=9\), so \(\log_{1/3}(9)=-2\). Also, \((1/3)^3=1/27\), so \(\log_{1/3}(1/27)=3\).
12. Estimate a Logarithm before Using Technology
When the argument is not an exact power, bracket it between nearby powers. To estimate \(\log_3(20)\), observe that
Because the base 3 is greater than 1,
Technology gives approximately \(2.727\), which is consistent with the bound. A displayed value outside the interval would signal an entry or base error.
For a base between 0 and 1, remember that the exponential function decreases. Since
the exponent still lies between 2 and 3: \(2<\log_{1/2}(0.20)<3\). The reversal occurs in the output inequalities of the exponential comparison, not in the final location between the two exponents.
13. Interpret the Common Logarithm
When no base is displayed, \(\log(x)\) means \(\log_{10}(x)\). It identifies the power of 10 needed to produce \(x\).
| Number | Common logarithm | Meaning |
|---|---|---|
| 10,000 | \(\log(10{,}000)=4\) | \(10^4=10{,}000\) |
| 0.01 | \(\log(0.01)=-2\) | \(10^{-2}=0.01\) |
| 1 | \(\log(1)=0\) | \(10^0=1\) |
Common logarithms also locate positive numbers by order of magnitude. If \(10^4<N<10^5\), then \(4<\log(N)<5\). Thus \(\log(62{,}000)\) must be between 4 and 5 before any calculator evaluation.
Do not confuse \(\log(100)\), which equals 2, with \(\log_{100}(10)\), which equals \(1/2\); changing the base changes the exponent question.
14. Read a Logarithmic Scale
Equal steps on a logarithmic scale represent equal multiplicative factors, not equal additive differences. On a base-10 logarithmic scale:
| Scale coordinate | Actual value | Change from previous mark |
|---|---|---|
| 0 | \(10^0=1\) | -- |
| 1 | \(10^1=10\) | Multiply by 10 |
| 2 | \(10^2=100\) | Multiply by 10 |
| 3 | \(10^3=1000\) | Multiply by 10 |
Therefore, 1 and 10 are one logarithmic unit apart, just as 100 and 1000 are one logarithmic unit apart. Their ordinary differences, 9 and 900, are irrelevant to the logarithmic spacing.
If two positive measurements differ by 2 units on a base-10 logarithmic scale, their actual values differ by a factor of \(10^2=100\). Moving downward by 3 units divides the actual value by \(10^3=1000\).
15. Common Errors
- Do not take a logarithm of a nonpositive real input.
- Treating a logarithm as ordinary division.
- Allowing zero or negative real arguments.
- Giving a numerical result without a domain check, units, or interpretation.
Key Takeaways
- Interpret logarithms as exponents and evaluate expressions using equivalent exponential statements.
- Core relationship: \(\log_b a=c\iff b^c=a\)
- Error check: Do not take a logarithm of a nonpositive real input.
- Evaluate \(\log_5(1/125)\).
- Rewrite \(7^x=20\) in logarithmic form.
- Explain why base 1 is invalid.