Manipulating Logarithmic Expressions
| English | Chinese | Pinyin |
|---|---|---|
| quotient rule | 除法法则 | chú fǎ fǎ zé |
| product rule | 乘法法则 | chéng fǎ fǎ zé |
| power rule | 幂法则 | mì fǎ zé |
| change-of-base | 换底 | huàn dǐ |
| equivalent | 等价 | děng jià |
Turning products into sums
- Before calculators, people multiplied huge numbers by adding their logarithms.
- That magic works because a log turns multiplication into addition.
- Three simple laws let you expand, condense, or re-base any log expression.
- They are just the exponent rules, viewed through the log's mirror.
Product and quotient rules
- The product rule 乘法法则 splits a product: $\log_b(xy) = \log_b x + \log_b y$.
- The quotient rule 除法法则 splits a quotient: $\log_b\!\left(\tfrac{x}{y}\right) = \log_b x - \log_b y$.
- Multiplication inside becomes addition outside; division becomes subtraction.
- These mirror "multiply powers → add exponents" and "divide → subtract".
The logarithm behind the laws
y = a·log(x − b) + c
These laws come from the exponent rules in reverse. Explore the log curve whose algebra they describe.
The product rule for logs says $\log_b(xy) =$
A log of a product is a sum of logs — the mirror of "multiplying powers adds exponents".
The quotient rule says $\log_b\!\left(\dfrac{x}{y}\right) = \log_b x - \log_b y$.
A log of a quotient is a difference of logs — the mirror of "dividing powers subtracts exponents".
The power rule
- The power rule 幂法则 brings an exponent out front: $\log_b(x^n) = n\log_b x$.
- So $\log_2(8^3) = 3\log_2 8 = 3 \cdot 3 = 9$.
- A power buried inside a log becomes a simple multiplier.
- Together, the three rules can fully expand or condense a log expression.
The power rule for logs says $\log_b(x^n) =$
A power inside a log comes out front as a multiplier: $\log_b(x^n) = n\log_b x$.
Change of base
- Calculators only have $\log$ (base 10) and $\ln$ (base $e$) buttons.
- The change-of-base 换底 formula handles any other base: $\log_b x = \dfrac{\log x}{\log b}$.
- So $\log_2 7 = \dfrac{\log 7}{\log 2} \approx 2.807$.
- Pick whichever base your tools support, then divide.
The ____ formula lets you compute $\log_2 7$ as $\dfrac{\log 7}{\log 2}$ on a calculator.
The change-of-base formula $\log_b x = \dfrac{\log x}{\log b}$ works with any convenient base.
Expand and condense
- Expanding uses the rules left-to-right to break one log into many.
- Condensing runs them right-to-left to combine many logs into one.
- Two expressions are equivalent 等价 if the rules convert one into the other.
- Choose the form that best fits the problem — solving usually wants a single log.
Select all correct log laws.
There is no rule for $\log(x+y)$ — the laws apply to products, quotients, and powers, not sums. The other three are correct.
There is no rule for the log of a sum: $\log(x + y)$ does not equal $\log x + \log y$. The laws only apply to products, quotients, and powers inside the log. Splitting a sum is one of the most common log mistakes.
Condense $2\log x + \log y - \log z$ into one logarithm.
- Power rule: $2\log x = \log x^2$.
- Product then quotient: $\log x^2 + \log y - \log z = \log\!\left(\dfrac{x^2 y}{z}\right)$.
- One clean logarithm, ready to solve or evaluate.
The log laws mirror the exponent rules: the product rule makes a sum, the quotient rule a difference, and the power rule a multiplier. The change-of-base formula evaluates any base with a calculator, and the rules show when two forms are equivalent — but never split $\log(x+y)$.