Exponential functions: graphs, domain, range, and asymptotes

Key idea

A basic exponential function has the form \[ f(x)=b^x \quad \text{where } b>0 \text{ and } b\neq 1. \] Core facts you should know:

• Domain: all real numbers \((-\infty,\infty)\).
• Range: \((0,\infty)\) because \(b^x\) is always positive.
• Key point: \(f(0)=b^0=1\), so the graph passes through \((0,1)\).
• Asymptote: the graph approaches \(y=0\) but never reaches it.
• Growth vs. decay: if \(b>1\) the function increases; if \(0<b<1\) it decreases.

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Summary

• For \(b^x\): domain \((-\infty,\infty)\), range \((0,\infty)\), asymptote \(y=0\).
• Growth if \(b>1\); decay if \(0<b<1\).

Solving exponential equations by rewriting bases and comparing exponents

Key idea

If you can rewrite both sides using the same base \(b\) (with \(b>0\), b≠ 1), then: \[ b^{A}=b^{B}\ \Rightarrow\ A=B. \] This is the fastest method when the right-hand side is a clean power of the base.

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Summary

• Rewrite both sides with the same base when possible.
• If \(b^{A}=b^{B}\), then \(A=B\) (for \(b>0\), b≠ 1).

Logarithms: definition, inverse relationship, and quick evaluation

Key idea

A logarithm answers this question: “What exponent gives this result?” \[ \log_b(x)=y \iff b^y=x \] with the important condition \(x>0\). Two common special bases:

• Common log: \(\log(x)=\log_{10}(x)\).
• Natural log: \(\ln(x)=\log_e(x)\).

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Summary

• \(\log_b(x)=y\) means \(b^y=x\) with \(x>0\).
• \(\log_{10}\) is the common log; \(\ln\) is the natural log (base \(e\)).

Log rules for simplifying, plus the change of base formula

Key idea

The three core log rules (for \(M>0\) and \(N>0\)) are:

• Product: \(\log_b(MN)=\log_b(M)+\log_b(N)\)
• Quotient: \(\log_b\!\left(\dfrac{M}{N}\right)=\log_b(M)-\log_b(N)\)
• Power: \(\log_b(M^p)=p\log_b(M)\)

If your calculator only has \(\ln\) and \(\log\), use change of base: \[ \log_b(a)=\frac{\ln a}{\ln b}=\frac{\log a}{\log b}. \]

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Summary

• Use product/quotient/power rules only when arguments are positive.
• Change of base: \(\log_b(a)=\dfrac{\ln a}{\ln b}\).

Solve logarithmic equations and check domains

Key idea

The simplest log equation has the form \(\log_b(\text{expression})=c\). Convert to exponential form: \[ \log_b(A)=c \iff A=b^c. \] Domain rule: every log argument must be positive (for example, \(x-1>0\)).

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Summary

• Convert \(\log_b(A)=c\) to \(A=b^c\).
• Always check the domain: every log argument must be \(>0\).

Use \(\ln\) to solve exponential equations when bases don’t match

Key idea

Logarithms are useful because they turn exponents into multiplication: \[ \ln\!\left(b^{g(x)}\right)=g(x)\ln(b). \] So when you can’t rewrite both sides with the same base, you can take \(\ln\) (or \(\log\)) of both sides and solve.

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Summary

• Taking \(\ln\) is powerful because \(\ln(b^{g(x)})=g(x)\ln(b)\).
• Use logs to solve exponentials when you can’t match bases easily.

Why exponential and logarithmic functions matter

Where exponentials and logs show up

• Exponential growth: populations, compound interest, viral spread.
• Exponential decay: half-life, depreciation, cooling models.
• Solving for time: logs help isolate exponents in models like \(P(t)=P_0b^t\) or \(P(t)=P_0e^{kt}\).
• Log scales: pH, decibels, earthquake magnitude (logarithmic scales compress large ranges).

Worked example: solve for the exponent

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Final recap

• Exponential functions: \(f(x)=b^x\) with \(b>0\), b≠ 1; domain \((-\infty,\infty)\), range \((0,\infty)\).
• Logs are inverses: \(\log_b(x)=y \iff b^y=x\) (with \(x>0\)).
• Common log: \(\log(x)=\log_{10}(x)\). Natural log: \(\ln(x)=\log_e(x)\).
• Log rules: product, quotient, and power rules (arguments must be positive).
• Change of base: \(\log_b(a)=\dfrac{\ln a}{\ln b}\).
• To solve exponentials: match bases when possible; otherwise take logs to bring down exponents.
• To solve logs: convert to exponential form and check the domain.