Logarithmic and Exponential Functions

Logs and exponentials carry substantial, predictable marks in 0606, and they’re a confidence topic: students who can manipulate logs stop fearing half the paper. The topic has four pillars: the laws, the $e$/$\ln$ pair, equation-solving, and the linear-form reduction that examiners adore.

The laws of logarithms (and their direction)

For the same base: $\log a + \log b = \log ab$, $\log a - \log b = \log(a/b)$, $n \log a = \log a^n$, plus $\log_a a = 1$, $\log_a 1 = 0$ and change of base $\log_b a = \lg a / \lg b$. All from the memorise list, never given.

The exam skill is choosing the direction: condensing many logs into one (to solve an equation) versus expanding one log into pieces (to use given values). Condensing is the right move 80% of the time: an equation with a single log on each side can have the logs dropped.

Solve $\log_2(x + 4) + \log_2(x - 3) = 3$ $\log_2[(x + 4)(x - 3)] = 3$ (M, laws applied) $(x + 4)(x - 3) = 2^3 = 8$ (M, log definition used) $x^2 + x - 20 = 0 \to (x + 5)(x - 4) = 0 \to x = -5$ or $x = 4$ $x = -5$ rejected ($x - 3$ would be negative) $\to$ $x = 4$ (A, with the rejection stated)

That rejection line is the topic’s signature mark, examiners flag it every session.

e, ln, and equations between them

$e^x$ and $\ln x$ are inverses; solving is “apply the other one”:

• $e^{2x-1} = 5 \to 2x - 1 = \ln 5 \to$ $x = (1 + \ln 5)/2$, exact form, exactly how Paper 1 wants it left
• $\ln(3x + 2) = 2 \to 3x + 2 = e^2 \to x = (e^2 - 2)/3$

For $a^x = b$ with awkward bases, take logs of both sides: $x = \lg b / \lg a$. Disguised quadratics are the favourite hard variant: $3^{2x} - 10\cdot3^x + 9 = 0 \to$ substitute $u = 3^x \to u^2 - 10u + 9 = 0 \to u = 1, 9 \to 3^x = 1$ or $9 \to$ $x = 0$ or $2$. The substitution line “let $u = 3^x$” is the method mark; finishing at $u$ and forgetting to return to $x$ is the classic dropped final mark.

The graphs

$y = e^x$: through $(0, 1)$, above the axis always, $x$-axis as asymptote. $y = \ln x$: through $(1, 0)$, defined only for $x > 0$, $y$-axis as asymptote, each the other’s reflection in $y = x$. Sketch marks attach to the labelled intercept and the asymptote behaviour. Transformed versions ($y = e^x + 3$, $y = \ln(x - 2)$) shift those features; track the asymptote, it’s where the B mark lives.

Reducing to linear form, the banker question

Experimental data follows $y = ax^n$ or $y = Ab^x$; taking logs straightens it:

• $y = ax^n$ $\to \log y = \log a + n \log x \to$ plot $\log y$ against $\log x$: gradient $n$, intercept $\log a$
• $y = Ab^x$ $\to \log y = \log A + x \log b \to$ plot $\log y$ against $x$: gradient $\log b$, intercept $\log A$

The exam routine: take logs of the model (M), match to $Y = mX + c$ stating what’s plotted (M), then extract the constants from the given gradient/intercept, remembering the intercept gives $\log a$, so $a = 10^{\text{intercept}}$ (the most-missed step). The same machinery appears from the graph side in straight-line graphs.

Common mistakes in this topic

• Inventing laws: $\log(a + b)$ is NOT $\log a + \log b$, the syllabus’s most-marked misconception
• Solutions to log equations left unchecked against positive-argument requirements
• Disguised quadratics solved to $u$ and abandoned
• Linear-form constants left as logs ($a = \log$-value instead of $10^{\text{value}}$)
• Decimal approximations on Paper 1 where $\ln 5$ should stand exact

Logs thread into calculus (derivative of $\ln x$, integrals producing $\ln$) and the linear-form work into coordinate geometry, the topic repays its drilling twice. It’s scheduled in week 3 of the revision plan.

If log manipulation still feels like spell-casting rather than algebra, one structured session usually flips it, free 1-hour trial with Teacher Rig via WhatsApp.