解题方法

Method 1: Standard Linearization

When to use

The equation is already linear or can be rearranged directly to $y = mx + c$ without applying logarithms.

Steps

1. Identify the target variables in the question — what is being plotted on $y$ and $x$ axes
2. Rearrange the given equation so that the $y$ variable is isolated on the LHS
3. Compare with $y = mx + c$ to identify $m$ (gradient) and $c$ (intercept)

通用公式

$\text{Given: } f(y) = A \cdot g(x) + B$

$\Rightarrow \text{gradient} = A,\quad \text{intercept} = B$

Formula

• Gradient = coefficient of the $x$-axis quantity
• Y-intercept = constant term after rearrangement

Mistakes to avoid

• Forgetting negative signs in the coefficient
• Not simplifying compound fractions before reading off coefficients

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Method 2: Log-Log Linearization

When to use

The equation is a power law: $y = ax^n$. The exponent $n$ is unknown and needs to be found from the gradient.

Steps

1. Take $\lg$ of both sides: $\lg y = \lg a + n \lg x$
2. Plot $\lg y$ on $y$-axis against $\lg x$ on $x$-axis
3. Gradient $= n$, y-intercept $= \lg a$
4. Find $a = 10^{\text{intercept}}$

何时选 $\lg$ 而非 $\ln$

• Power laws $y = ax^n$ → use $\lg$ (base 10)
• Exponential $y = ae^{kx}$ → use $\ln$ (base $e$)

Formula

$\lg y = \underbrace{n}_{\text{gradient}} \lg x + \underbrace{\lg a}_{\text{intercept}}$

Mistakes to avoid

• Using $\ln$ instead of $\lg$ (they give different gradient values for power laws)
• Writing $\lg y$ without dividing by units: write $\lg(y/\text{unit})$
• Forgetting that $a = 10^{\text{intercept}}$, not just the intercept itself

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Method 3: Log-Linear Linearization

When to use

The equation is exponential: $y = ae^{kx}$.

Steps

1. Take $\ln$ of both sides: $\ln y = \ln a + kx$
2. Plot $\ln y$ on $y$-axis against $x$ on $x$-axis
3. Gradient $= k$, y-intercept $= \ln a$
4. Find $a = e^{\text{intercept}}$

Formula

$\ln y = \underbrace{k}_{\text{gradient}} x + \underbrace{\ln a}_{\text{intercept}}$

Mistakes to avoid

• Plotting $\ln y$ against $\ln x$ instead of against $x$
• Sign errors: if $y = ae^{-kx}$, then gradient $= -k$
• Forgetting units: $\ln(V/\text{V})$ not $\ln V$

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Method 4: Reciprocal Linearization

When to use

The equation has the form $y = a + \frac{b}{x}$ or can be rearranged to show an inverse relationship.

Steps

1. Rearrange to $y = a + b \cdot \frac{1}{x}$
2. Plot $y$ on $y$-axis against $\frac{1}{x}$ on $x$-axis
3. Gradient $= b$, y-intercept $= a$

Formula

$y = \underbrace{b}_{\text{gradient}} \cdot \frac{1}{x} + \underbrace{a}_{\text{intercept}}$

Mistakes to avoid

• Plotting $y$ against $x$ instead of $y$ against $1/x$
• Forgetting that the intercept is $a$, not $b$
• Not checking whether the relationship passes through the origin

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方法选择速查

看见...	用 Method	作图方式
$y = mx + c$	Standard	$y$ vs $x$
$y = ax^n$	Log-log	$\lg y$ vs $\lg x$
$y = ae^{kx}$	Log-linear	$\ln y$ vs $x$
$y = a + b/x$	Reciprocal	$y$ vs $1/x$
$x$ and $y$ are both powers	Log-log	$\lg y$ vs $\lg x$
One variable is in exponent	Log-linear	$\ln y$ vs $x$