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A Logarithm Is the Power

\begin{gathered}x = a^b \;\Rightarrow\; \log_a x = b\\[4pt] \log_{\textcolor{#b45309}{a}} x = b \;\Rightarrow\; x = \textcolor{#b45309}{a}^b\end{gathered}

A logarithm is a power: $\log_2 7$ is the power you put on $2$ to get $7$ . Converting between log and indices keeps the base: the base of the log is the base of the indices — $b$ moves up to become the power. Reflex: Calculate $\log_2 7$ as $\frac{\log 7}{\log 2}$ .

Must-Know Laws of Logarithms

\begin{aligned} \log_a xy &= \log_a x + \log_a y && \text{(add power)} \\ \log_a \tfrac{x}{y} &= \log_a x - \log_a y && \text{(minus power)} \\ \log_a x^r &= r\log_a x && \text{(bring down power)} \\ \log_a 1 &= 0, \ \ \log_a a = 1 \\ \log_a b &= \frac{\log_c b}{\log_c a} && \text{(change of base)} \\ \log_a b &= \frac{1}{\log_b a} \end{aligned}

Same base → combine freely. Different bases → change of base first. Reflex: Check bases match before combining; if not, change of base first.

Fake Log Laws: Don't Combine Whole Logs

\begin{gathered}(\log_a x)(\log_a y) \neq \log_a(xy)\\[3pt] \frac{\log_a x}{\log_a y} \neq \log_a\!\left(\frac{x}{y}\right)\\[3pt] (\log_a x)^n \neq n\log_a x\end{gathered}

Log laws act on the argument, not on whole logs — $(\log_a x)^n$ stays as is. Reflex: Bring down the power only when it sits on the argument.

lg and ln: The Shorthand Bases

\begin{gathered}\lg 7 = \log_{10} 7, \ \ \lg 10 = 1\\[3pt] \ln 7 = \log_e 7, \ \ \ln e = 1\end{gathered}

$\lg$ is $\log_{10}$ , $\ln$ is $\log_e$ . Reflex: See $\lg$ with a 10, or $\ln$ with an $e$ — it simplifies to 1.

Rewrite, Then Compare Arguments

\log_a(\;\;) = \log_a(\;\;)

Rewrite the equation into this form using log laws, then compare arguments directly. Reflex: Check every log argument stays positive — reject answers that fail this.

Log ⇔ Index: Swap Both Ways

y = a^x \iff \log_a y = x

Works either direction: $7=2^x \Rightarrow x=\log_2 7$ , or $\log_3 x=5 \Rightarrow x=3^5$ . Reflex: Unknown trapped inside a log? Convert to indices to free it.

Same Base: Compare Powers. Different Bases: Take Logs

\begin{gathered}\text{Same base — compare powers}\\[3pt] \begin{aligned}5^{x+2} &= 25^{x-1}\\ 5^{x+2} &= 5^{2x-2}\\ x+2 &= 2x-2\\ x &= 4\end{aligned}\end{gathered}

\begin{gathered}\text{Different bases — take logs}\\[3pt] \begin{aligned}3^x &= 7\\ \lg 3^x &= \lg 7\\ x\lg 3 &= \lg 7\\ x &= \tfrac{\lg 7}{\lg 3}\end{aligned}\end{gathered}

Match the bases when you can, then compare powers. If the bases can never match, take $\lg$ of both sides and bring the power down. Reflex: Treat $\lg 7$ , $\lg 5$ as plain numbers — collect like algebra, then calculator.

Given p = logₐx: Convert Everything to Base a

\begin{gathered}\log_b M = \frac{\log_a M}{\log_a b}\\[3pt] \log_b a = \frac{1}{\log_a b}\end{gathered}

Rewrite every asked log in base $a$ until $\log_a x$ appears, then swap it for $p$ . Reflex: Change to the base of the given variable, and turn every number into a power of that base.

The Logarithmic Curve: Slowing Growth

y = \log_a x

All bases ( $\log_2 x$ , $\lg x$ , $\ln x$ ) share this shape — growth slows as $x$ increases. Reflex: Remember the $y$ -axis is an asymptote, and the curve always passes through $(1, 0)$ .

Minus Signs Flip the Log Graph

Start from $\ln x$ : minus in front flips in the $x$ -axis; minus in front of $x$ flips in the $y$ -axis; both together flip twice. Each curve gets closer and closer to the $y$ -axis — never touching it. Reflex: Learn the SHAPE — spot which side the minus sits on.

Log Shift Sketch: Shape Stays, Asymptote Moves

\begin{gathered}y = \lg(2-x)\\[2pt] \text{shape of } \lg(-x), \ \text{asymptote } x = 2\end{gathered}

Shape ← structure of the equation.

Asymptote ← set inside of log to 0.

Numbers and base don't affect shape. Reflex: Set the inside of the log to zero for the asymptote; read structure alone for the shape.

Exponential Models: Long-Run Behaviour

\text{As } t \to \infty, \ e^{-kt} \to 0 \quad (k > 0)

Initial value: sub $t=0$ . Solving for time: isolate exponential, take lg/ln. Long run: let $t\to\infty$ — the arrow reads "approaches". Reflex: Graph only $t \geq 0$ , label initial value on vertical axis.

Solving Equations Graphically

e^x = x + 2

You already have the curve $y = e^x$ . To solve $e^x = x + 2$ , draw the line $y = x + 2$ on the same axes. Every crossing point is one solution — read off its $x$ -value. The number of crossings is the number of solutions. Reflex: Always name the added graph with its own " $y=$ " — draw $y = x + 2$ , not just $x + 2$ .