Unit 11 · Lesson 2 of 5
Logarithms
You'll be able to: switch between log form and exponential form, and find exact values like \(\log_2 8\), \(\log 1000\), and \(\ln e^3\).
Step 1 of 5Warm-up
Two quick questions from earlier lessons. Answer from memory first, then check.
\(f(x) = 2 \cdot 3^x\). Find \(f(3)\).
Power first: \(3^3 = 27\). Then \(2 \cdot 27 = 54\). Answer: \(54\).
\(8^{2/3}\)
The bottom of the exponent is the root: \(\sqrt[3]{8} = 2\). The top is the power: \(2^2 = 4\). Answer: \(4\).
Step 2 of 5The idea
A logarithm answers one question: what exponent do I need? For example, \(\log_2 8\) asks "2 to what power gives 8?" Since \(2^3 = 8\), \(\log_2 8 = 3\).
Definition. \(\log_b x = y\) means \(b^y = x\). The little number \(b\) is the base of the power in both forms.
Two special logs. \(\log x\) with no base written means base 10 (the common log). \(\ln x\) means base \(e\) (the natural log).
Handy facts. \(\log_b 1 = 0\), \(\log_b b = 1\), and \(\log_b\left(b^k\right) = k\).
Domain. You can only take the log of a positive number: \(\log_b x\) needs \(x \gt 0\).
Why only positive numbers? A positive base raised to any power is always positive, so no exponent can ever give 0 or a negative number.
The log is the inverse of the exponential (Lesson 0053): \(y = \log_2 x\) undoes \(y = 2^x\). Its graph is the mirror image over the line \(y = x\).
Why this shows up on the placement test: every log equation and log property starts with switching between \(\log_b x = y\) and \(b^y = x\).
Step 3 of 5Worked examples
Example 1: switch forms
- \(\log_2 32 = 5\) becomes \(2^5 = 32\). The base 2 is raised to the answer 5.
- \(10^{-2} = 0.01\) becomes \(\log 0.01 = -2\). Base 10, so the base is not written.
- \(e^0 = 1\) becomes \(\ln 1 = 0\). Base \(e\) is written as ln.
Example 2: find \(\log_3 \dfrac{1}{9}\) and \(\ln e^3\)
- Ask: 3 to what power is \(\dfrac19\)?
- \(9 = 3^2\), so \(\dfrac19 = \dfrac{1}{3^2} = 3^{-2}\). A negative exponent means "one over".
- So \(\log_3 \dfrac19 = -2\).
- For \(\ln e^3\), ask: \(e\) to what power is \(e^3\)? The answer is right there: \(\ln e^3 = 3\).
Step 4 of 5Practice
Type your answer and press Check. If you miss twice, open the worked solution. Then mark how it went. No calculator needed.
\(\log_2 16\)
2 to what power is 16? \(2^4 = 16\). Answer: \(4\).
\(\log 1000\)
No base written means base 10. 10 to what power is 1000? \(10^3 = 1000\). Answer: \(3\).
\(\log_5 \dfrac{1}{25}\)
\(25 = 5^2\), so \(\dfrac{1}{25} = 5^{-2}\). Answer: \(-2\).
\(\ln e^5\)
\(\ln\) means base \(e\). \(e\) to what power is \(e^5\)? Answer: \(5\).
\(\log_4 2\)
4 to what power is 2? Since \(\sqrt{4} = 2\), and a square root is the power \(\dfrac12\), \(4^{1/2} = 2\).
Answer: \(\dfrac12\).
Which is the exponential form of \(\log_7 49 = 2\)?
- \(2^7 = 49\)
- \(49^2 = 7\)
- \(7^2 = 49\)
- \(7^{49} = 2\)
\(\log_b x = y\) means \(b^y = x\). Here \(b = 7\), \(y = 2\), \(x = 49\), so \(7^2 = 49\). Choice C.
Check: \(7^2\) really is 49. The others are false (\(2^7 = 128\), \(49^2 = 2401\)).
What is the domain of \(f(x) = \log(x - 3)\)? Give it in interval notation.
The input of a log must be positive: \(x - 3 \gt 0\).
Add 3: \(x \gt 3\). In interval notation: \((3, \infty)\).
\(\log_3 27 - \log_2 \dfrac{1}{4}\)
\(\log_3 27 = 3\), because \(3^3 = 27\).
\(\log_2 \dfrac14 = -2\), because \(\dfrac14 = 2^{-2}\).
\(3 - (-2) = 3 + 2 = 5\). Answer: \(5\).
Step 5 of 5Wrap-up
You can now read any log as a question about an exponent, switch between \(\log_b x = y\) and \(b^y = x\), and spot that only positive numbers have logs.
Read or watch next: OpenStax Intermediate Algebra 2e, 10.3 Evaluate and Graph Logarithmic Functions (free textbook, with more practice at the end of the section).
Also good: Khan Academy: Introduction to logarithms (short videos). Quick rules: Exponentials and logs cheat sheet.
Something unclear? Ask your teacher bot a follow-up question, for example: "Why is the log of a negative number undefined?" It knows this course and can explain things another way.