Exponential & Logarithmic Functions

An exponential function has the form

The variable lives in the exponent. Here $a$ is the initial value (the $y$-intercept, since $b^0 = 1$ gives $y=a$), and $b$ is the base or growth factor.

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• If $b > 1$: exponential growth (curve rises to the right).
• If $0 < b < 1$: exponential decay (curve falls to the right).

For $a>0$: domain is all real numbers; range is $y > 0$; the line $y = 0$ is a horizontal asymptote the curve approaches but never touches.

Starting from the parent $y = b^x$, the function $y = a\,b^{\,x-h} + k$ is:

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• shifted right $h$ and up $k$;
• stretched by $|a|$ (reflected over the $x$-axis if $a<0$);
• its horizontal asymptote moves to $y = k$, and the range becomes $y > k$ (or $y<k$ if $a<0$).

The constant $e \approx 2.71828$ is the natural base of growth. It arises as the limit of $\left(1+\tfrac{1}{n}\right)^n$ as $n \to \infty$. The function $y = e^x$ is a growth curve just like any $b^x$ with $b>1$; it models continuous change.

For principal $P$, annual rate $r$ (as a decimal), and time $t$ in years:

Use $n=1$ (annually), $4$ (quarterly), $12$ (monthly), $365$ (daily).

A logarithm answers the question “to what power?” The definition is the inverse relationship

So $\log_b y$ is the exponent you put on $b$ to get $y$. To evaluate, rewrite in exponential form.

Two bases are so useful they get their own notation:

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• Common log: $\log x$ means $\log_{10} x$ (base 10).
• Natural log: $\ln x$ means $\log_e x$ (base $e$).

Inverse facts: $\log 10^k = k$, $\ln e^k = k$, $10^{\log x}=x$, $e^{\ln x}=x$.

Because $\log_b x$ undoes $b^x$, the graph of $y = \log_b x$ is the reflection of $y = b^x$ across the line $y = x$. Consequences for $y=\log_b x$ (with $b>1$):

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• Domain $x > 0$; range all real numbers.
• Vertical asymptote at $x = 0$ (the $y$-axis).
• passes through $(1,0)$ since $\log_b 1 = 0$.

A shift $y = \log_b(x-h)$ moves the asymptote to $x = h$ and the domain to $x > h$.

These convert products into sums and exponents into multipliers. Reading them left-to-right expands; right-to-left condenses.

To evaluate a log in any base with a calculator:

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• Common base: rewrite both sides with the same base, then set the exponents equal.
• Take a log: if a common base is awkward, isolate the power and apply $\log$ or $\ln$ to both sides, then use the power rule to bring the exponent down.

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• Use log laws to condense into a single log if needed.
• Rewrite in exponential form, or use “one-to-one” ($\log_b M = \log_b N \Rightarrow M = N$).
• Solve the resulting equation.
• Check every answer: the input of any log must be positive. Discard extraneous solutions.

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• Growth / decay: $A = A_0(1 \pm r)^t$ or continuous $A = A_0 e^{kt}$.
• Half-life: $A = A_0\left(\tfrac{1}{2}\right)^{t/h}$, where $h$ is the half-life.
• pH: $\text{pH} = -\log[\mathrm{H}^+]$, where $[\mathrm{H}^+]$ is the hydrogen-ion concentration.
• Richter magnitude: $M = \log\!\dfrac{I}{I_0}$; each whole number is a $\times 10$ jump in intensity.

Change of base can be written in any common base $c$:

This is the key to telescoping a product of logs. Watch a chain of $\log_k(k{+}1)$ terms collapse when every log is rewritten over the same base $c$:

Each numerator cancels the next denominator, leaving $\dfrac{\log n}{\log 2} = \log_2 n$. The whole chain equals a single log.

When a problem fixes one log as a letter, express others by breaking numbers into prime factors and applying the three laws. If $\log_2 3 = a$, then:

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• $\log_2 6 = \log_2(2\cdot 3) = 1 + a$;
• $\log_2 12 = \log_2(2^2\cdot 3) = 2 + a$;
• $\log_2 \tfrac{9}{8} = \log_2 3^2 - \log_2 2^3 = 2a - 3$;
• $\log_3 2 = \dfrac{1}{\log_2 3} = \dfrac{1}{a}$  (reciprocal / change of base).

An equation with terms in $2^x$ and $2^{2x}$ (or $e^x$ and $e^{2x}$) is secretly a quadratic. Substitute $u = 2^x > 0$, using $2^{2x} = (2^x)^2 = u^2$, solve the quadratic, then back-substitute. Reject any $u \le 0$, since $2^x$ is always positive.

When the variable appears in both the base and the exponent, as in $x^{\log x}$, take $\log$ of both sides first; the power rule turns the exponent into a factor and creates a quadratic in $\log x$. Let $t = \log x$.

A system mixing sums and products of logs is solved by condensing each equation, then converting to ordinary algebra. Keep the same base throughout, and enforce the domain (every argument $>0$) at the end.

Two continuous models worth knowing beyond $A = Pe^{rt}$:

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• Doubling / tripling time: set $A = 2P$ in $Pe^{rt}$ to get $t = \dfrac{\ln 2}{r}$ (the “rule of $69$” in disguise).
• Logistic growth: real populations cannot grow forever; they level off at a carrying capacity $L$: @@BLOCK0@@ Early on it looks exponential; as $t\to\infty$, $e^{-kt}\to 0$ so $P\to L$. The curve is S-shaped (sigmoidal) with a horizontal asymptote at $P = L$.

Solving an inequality adds a direction rule on top of the equation techniques:

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• Because $\log_b x$ (with $b>1$) is increasing, applying it preserves the inequality: $M < N \Leftrightarrow \log_b M < \log_b N$ for $M,N>0$.
• If the base satisfies $0 < b < 1$, both $b^x$ and $\log_b x$ are decreasing, so the inequality flips.
• Always intersect your answer with the domain (arguments of every log must be positive).