Exponential and Logarithmic Equations
| English | Chinese | Pinyin |
|---|---|---|
| exponential equation | 指数方程 | zhǐ shù fāng chéng |
| logarithmic equation | 对数方程 | duì shù fāng chéng |
| extraneous | 增根 | zēng gēn |
Finding a hidden exponent
- "$2$ to what power equals $5$?" is an equation with the unknown up in the exponent.
- You cannot just divide it out — the variable is trapped inside a power.
- Logarithms are the tool that frees it, and exponential form frees log equations.
- Between these two moves, you can solve almost any exponential or log equation.
Solving exponential equations
- An exponential equation 指数方程 has the variable in an exponent, like $2^x = 8$.
- If both sides can share a base, match them: $2^x = 2^3$ gives $x = 3$.
- If the bases will not match, take a logarithm of both sides.
- The power rule then brings the exponent down where you can solve for it.
Find where the exponential reaches a target value
y = 2ˣ
Solving 2ˣ = 5 means finding the x where this curve hits height 5. A logarithm gives that x exactly.
Solve $2^x = 8$ by matching bases.
$8 = 2^3$, so $2^x = 2^3$ gives $x = 3$ — equal bases mean equal exponents.
To solve $2^x = 5$, where the bases will not match, you should…
Taking $\log$ of both sides brings the exponent down: $x = \log_2 5 \approx 2.32$.
Solving logarithmic equations
- A logarithmic equation 对数方程 has the variable inside a log, like $\log_2 x = 4$.
- Rewrite it in exponential form: $\log_2 x = 4$ means $x = 2^4 = 16$.
- If several logs appear, condense them into one first, then convert.
- Exponential form is the master key for any single-log equation.
Solve the logarithmic equation $\log_2 x = 4$.
Rewrite in exponential form: $x = 2^4 = 16$.
Check for extraneous solutions
- Solving can produce a candidate that breaks the original equation's domain.
- A logarithm needs a positive argument, so any solution giving $\log(\text{negative})$ is rejected.
- Such a false answer is called an extraneous 增根 solution.
- Always substitute back and confirm every argument stays positive.
A candidate solution that makes a log argument negative must be rejected as ____.
An extraneous solution satisfies the rearranged equation but breaks the original domain — always check.
Select all true statements about solving these equations.
Some candidates are extraneous and must be rejected. The other three are sound strategies.
Inequalities
- Exponential and logarithmic functions are monotonic — always increasing or always decreasing.
- So an inequality like $2^x > 8$ solves the same way, keeping the direction: $x > 3$.
- For a decreasing base ($0 < b < 1$), the inequality sign flips.
- Solve as an equation first, then decide which side of the boundary satisfies it.
Never forget to check log-equation answers. Solving $\log(x) + \log(x-3) = 1$ may hand you $x = 5$ and $x = -2$, but $x = -2$ makes $\log(-2)$ undefined — it is extraneous and must be thrown out.
Solve $2^x = 5$.
- Bases will not match, so take $\log_2$ of both sides.
- $\log_2(2^x) = \log_2 5$, and the left side simplifies to $x$.
- So $x = \log_2 5 \approx 2.32$ — the exponent the logarithm revealed.
Solve an exponential equation by matching bases, or by taking a logarithm to bring the exponent down. Solve a logarithmic equation by rewriting it in exponential form. Always check for extraneous solutions, since a log's argument must stay positive.