Logarithms
Introduced| English |
|---|
| exponential/ˌekspəˈnenʃl/ |
| logarithm/ˈlɒɡərɪθəm/ |
| base/beɪs/ |
| natural log/ˈnætʃərəl lɒɡ/ |
| common log/ˈkɒmən lɒɡ/ |
The reverse question
- Exponentials answer "start at 1 and multiply by 2 five times — what do you get?" ($2^5 = 32$).
- Logarithms answer the reverse: "how many times must I multiply by 2 to reach 32?" ($5$).
- That reverse question comes up everywhere: pH, earthquakes, decibels, compound interest.
- A logarithm is simply the exponent, pulled back out.
A logarithm is an exponent
- A logarithm 对数 answers: to what power must the base 底数 be raised to get a value?
- $\log_b y = x$ means exactly $b^x = y$ — the log is the exponent $x$.
- So $\log_2 8 = 3$ because $2^3 = 8$.
- Reading a log as "the exponent that works" makes it far less mysterious.
Read a logarithm off the exponential curve
y = 2ˣ
Find the y-value 8 on the curve y = 2ˣ. The x that produces it is the exponent — that is log base 2 of 8.
The statement $\log_b y = x$ means the same as…
A logarithm is the exponent: $\log_b y = x$ asks "$b$ to what power gives $y$?", so $b^x = y$.
Evaluate $\log_2 8$.
$2^3 = 8$, so the exponent is $3$: $\log_2 8 = 3$.
Common and natural logs
- Two bases are so common they get shortcuts.
- The common log 常用对数 $\log$ (with no base shown) has base $10$.
- The natural log 自然对数 $\ln$ has base $e \approx 2.718$.
- Calculators have both buttons; other bases use the change-of-base formula.
The natural log $\ln$ uses which base?
The natural log has base $e$; the common log ($\log$ with no base written) has base $10$.
Select all true statements about logarithms.
The argument of a log must be positive, so you cannot take the log of a negative. The other three are correct.
Converting between forms
- Every exponential 指数 statement has a matching logarithmic one.
- $b^x = y \iff \log_b y = x$ — the same fact written two ways.
- Rewrite $10^3 = 1000$ as $\log_{10} 1000 = 3$.
- Switching forms is the key trick for solving equations with unknown exponents.
The argument must be positive
- You can only take the logarithm of a positive number.
- A positive base raised to any power stays positive, so $b^x$ never reaches $0$ or a negative.
- Therefore $\log_b y$ is undefined for $y \le 0$.
- This restriction shapes the domain of every logarithmic function.
The argument (input) of a logarithm must be ____, because a positive base to any power stays positive.
Since $b^x > 0$ always, no exponent can produce zero or a negative — so $\log_b y$ needs $y > 0$.
$\log_b y$ is the exponent, not a product. $\log_2 8$ is $3$ (the power), not $2 \times 8$. And you can never take the log of $0$ or a negative number — the argument must be strictly positive.
Solve $2^x = 16$ using a logarithm.
- Rewrite in log form: $x = \log_2 16$.
- Ask "$2$ to what power is $16$?" — since $2^4 = 16$, the answer is $4$.
- So $x = 4$: the logarithm pulled the exponent out for us.
A logarithm is an exponent: $\log_b y = x$ means $b^x = y$. The common log uses base $10$ and the natural log uses base $e$. Every exponential statement converts to a log statement, and the argument of a log must always be positive.