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)

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

Quotient Rule (logs)

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

Power Rule (logs)

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

No “sum rule” for logs

In general, $(\log_b(M+N) \ne \log_b(M)+\log_b(N))$; 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).