Unit 2 Notes: Understanding Logarithms (AP Precalculus)

Logarithm (log)

A function that answers “what exponent do I need?”; it reverses exponentiation.

Definition of a logarithm

(\log_b(x)=y) if and only if (b^y=x).

Base restrictions for logarithms

For real logarithms, the base must satisfy (b>0) and (b\ne 1).

Argument (input) restriction for logarithms

For real logarithms, the argument must be positive: (x>0).

Logarithmic expression

Any algebraic expression involving logarithms, e.g., (\log_3(7x-1)) or (2\log(x)-\log(5)).

Common logarithm

(\log(x)), typically meaning logarithm base 10.

Natural logarithm

(\ln(x)), the logarithm base (e).

Constant (e)

A special irrational constant (~2.71828) used as the base of the natural logarithm and common in continuous growth models.

Inverse relationship (logs and exponentials)

(y=b^x) and (y=\log_b(x)) are inverses; each “undoes” the other.

Domain and range of (y=b^x)

Domain: all real numbers; Range: (y>0).

Domain and range of (y=\log_b(x))

Domain: (x>0); Range: all real numbers.

Monotonicity of (y=\log_b(x))

Increasing if (b>1); decreasing if (0<b<1).

Anchor point: (\log_b(1)=0)

Because (b^0=1), every log graph passes through ((1,0)).

Anchor point: (\log_b(b)=1)

Because (b^1=b), (\log_b(b)=1).

Undoing identity (log of an exponential)

(\log_b(b^x)=x) for valid base (b).

Undoing identity (exponential of a log)

(b^{\log_b(x)}=x) for (x>0) and valid base (b).

Product Rule (logs)

(\logb(MN)=\logb(M)+\log_b(N)), with (M>0), (N>0).

Quotient Rule (logs)

(\logb\left(\frac{M}{N}\right)=\logb(M)-\log_b(N)), with (M>0), (N>0).

Power Rule (logs)

(\logb(M^p)=p\logb(M)), with (M>0).

No “sum rule” for logs

In general, (\logb(M+N)\ne \logb(M)+\log_b(N)); logs do not distribute over addition.

Change of base formula

(\log_b(x)=\frac{\log(x)}{\log(b)}=\frac{\ln(x)}{\ln(b)}), allowing computation with base 10 or base (e).

Condense (logarithms)

Combine multiple log terms into a single logarithm by reversing the product/quotient/power rules (often used to solve equations).

Expand (logarithms)

Rewrite one logarithm as a sum/difference by applying product/quotient/power rules (often used to simplify or match forms).

Extraneous solution (log equations)

A solution produced by algebra that must be rejected because it makes a log argument nonpositive, violating the domain restriction.

Vertical asymptote of a logarithmic function

For (y=\logb(x)), the vertical asymptote is (x=0); for (y=a\logb(x-h)+k), it shifts to (x=h).