Lesson 2-2: Logarithmic Functions – Properties & Transformations

Learning Objectives

Connect logarithms to exponentials via inverse relationships and graph symmetry.
Determine domains/ranges and interpret log scales (base 10, base e).
Apply product, quotient, and power identities to simplify expressions.

Use change-of-base to evaluate non-standard logs and compare magnitudes.
Perform graph transformations (shifts, stretches) and identify asymptotes.

Logarithms as Inverses

The logarithm base $a$ is the inverse of the exponential base $a$ . If $y=a^x$ , then $x=\log_a y$ with domain $y>0$ .

\log_a a^x = x, quad a^{\log_a x}=x, quad a>0, a\neq 1, x>0

Graphically, the log graph is the reflection of the exponential graph across the line $y=x$ . The domain of $\log_a x$ is $(0,\infty)$ , range is $(-\infty,\infty)$ , and there is a vertical asymptote at $x=0$ .

Core Identities

Product Rule

\log_a(MN) = \log_a M + \log_a N

Combine factors as sums

Quotient Rule

\log_a(\tfrac{M}{N}) = \log_a M - \log_a N

Differences correspond to ratios

Power Rule

\log_a(M^k) = k\, \log_a M

Exponents become coefficients

Inverse

a^{\log_a x} = x, \; \log_a(a^x)=x

Undoing via inverse operations

Change of Base

\log_a b = \tfrac{\log_c b}{\log_c a}

Switch to convenient base (10 or e)

Change of Base & Practical Evaluation

To compute logs with arbitrary bases using a calculator offering $ln$ or $log_{10}$ :

\log_a b = \tfrac{\ln b}{\ln a} = \tfrac{\log_{10} b}{\log_{10} a}

This also simplifies comparisons of magnitudes and converts multiplicative relationships into additive ones for analysis.

Transformations & Asymptotes

Vertical & Horizontal Shifts

y=\log_a(x-h)+k

Domain shifts to $x>h$ , vertical asymptote at $x=h$ .

Scale Changes

y=A\,\log_a(bx)

Vertical stretch by $A$ , horizontal scaling by $1/b$ .

Worked Examples

Example 1: Simplify

$\log_2(8\cdot 4^2)-2\log_2\sqrt{2}+\log_2 1$

Show Solution

Use product and power rules: $=\log_2 8 + 2\log_2 4 - 2\log_2 2^{1/2} + 0$

$=3 + 2\cdot 2 - 2\cdot(1/2)$ = 3 + 4 - 1 = 6.

Example 2: Evaluate without Calculator

$\log_5 12 \cdot \log_{12} 25$

Show Solution

$=\tfrac{\ln 12}{\ln 5}\cdot \tfrac{\ln 25}{\ln 12} = \tfrac{\ln 25}{\ln 5}$

$= \tfrac{\ln 5^2}{\ln 5}=2$ .

Example 3: Find Domain & Asymptote

For $y=\log_3(x-2)+4$ , find the domain and vertical asymptote.

Show Solution

Require $x-2>0$ ⇒ $x>2$ . Asymptote: $x=2$ .

Example 4: Expand and Condense

Expand $\log_b\left(\tfrac{a^3 \sqrt{c}}{d^2}\right)$ and then condense back.

Show Solution

$=3\log_b a + \tfrac{1}{2}\log_b c - 2\log_b d$

Condense: $= \log_b \left( \tfrac{a^3 c^{1/2}}{d^2} \right)$

Practice Problems

Problem 1

Write the domain of $y=\ln(2x-5)$ and state its vertical asymptote.

Show Solution

$2x-5>0\Rightarrow x>2.5$ ; asymptote $x=2.5$ .

Problem 2

Use identities to simplify $\log_7(49x^2) - \log_7(7x)$ .

Show Solution

$= \log_7\left(\tfrac{49x^2}{7x}\right)=\log_7(7x)=1+\log_7 x$

Problem 3

Evaluate $\log_3 7$ to three decimals using change-of-base.

Show Solution

$=\tfrac{\ln 7}{\ln 3}$ ≈ 1.771.

Problem 4

Expand $\ln\left( \tfrac{e^{3x} \sqrt{1+x}}{4} \right)$ .

Show Solution

$=3x + \tfrac{1}{2}\ln(1+x) - \ln 4$

Problem 5

Condense $2\log_b x - \tfrac{1}{2}\log_b y + \log_b z$ into a single logarithm.

Show Solution

$= \log_b \left( \tfrac{x^2 z}{\sqrt{y}} \right)$

Key Takeaways

Logs are defined only for positive arguments and invert exponentials.
Identities turn multiplicative structures into additive ones; beware of sums.
Change-of-base enables evaluation and comparison using ln or log base 10.
Shifts move the vertical asymptote; scales stretch graphs without changing asymptote nature.

Log Scales & Applications

Decibels

Sound level: $L=10log_{10}(I/I_0)$ , comparing intensity to reference.

pH Scale

Acidity: $mathrm{pH}=-log_{10}[H^+]$ , each unit tenfold change in hydrogen ion concentration.

Practice Bank

Bank 1: Expand Products

Apply product rule to separate factors.
Simplify coefficients and constants.
State domain constraints explicitly.

Bank 2: Condense Sums

Turn sums/differences into a single log.
Check power coefficients become exponents.
Verify positivity of arguments.

Bank 3: Change of Base

Evaluate with ln or log10 via change-of-base.
Compare magnitudes of unfamiliar bases.
Explain numerical rounding impact.

Bank 4: Domain Analysis

Solve inequalities ensuring arguments > 0.
Express domain in interval notation.
Relate to vertical asymptotes.

Bank 5: Transformations

Apply shifts and scales to $log_a(x)$ .
Find new asymptote and intercepts.
Sketch accurately with labeled axes.

Bank 6: Mixed Bases

Convert to a common base strategically.
Simplify expressions before evaluating.
State rounding and units if any.

Bank 7: Identity Pitfalls

Explain why $log(M+N)$ ≠ $log M + log N$ .
Give a counterexample numerically.
State correct alternative manipulations.

Bank 8: Composition with Exp

Use $log_a a^x=x$ to simplify.
Handle domains for $a^{log_a x}$ .
Relate to inverse functions.

Bank 9: Real-World Scales

Apply dB and pH formulas.
Interpret a one-unit change in context.
Compute comparative changes.

Bank 10: Log Inequalities

Transform to base-exponential inequalities.
Track monotonicity with base conditions.
Present solution sets clearly.

Bank 11: Parameter Effects

Vary base a and observe graph changes.
Explain slope comparison on log scales.
Connect to sensitivity analysis.

Bank 12: Numeric Stability

Avoid subtractive cancellation in logs.
Prefer identities that improve conditioning.
Estimate rounding error impact.

Bank 13: Piecewise Domains

Partition domains for composite logs.
State asymptotes on each interval.
Sketch complete graphs.

Bank 14: Modeling with Logs

Linearize multiplicative models.
Interpret slope/intercept physically.
Check residuals for nonlinearity.

Bank 15: Composition & Inverses

Show inverse relations algebraically.
Confirm by composing functions.
Discuss domain/range swaps.

Bank 16: Challenge Mix

Combine product, quotient, power rules.
Convert bases mid-simplification.
Verify final domain precisely.

FAQ (Extended)

Q: Why can't log distribute over addition?

Because $\log(M+N)$ ≠ $\log M + \log N$ . Only products, quotients, and powers obey log identities.

Q: When to choose ln vs log10?

Use ln for mathematical derivations and continuous models; use log10 for orders-of-magnitude contexts in engineering/science.

Back to Lesson 2-1 Next Lesson 2-3

Logarithmic Functions: Properties & Transformations

Learning Objectives

Logarithms as Inverses

Core Identities

Product Rule

Quotient Rule

Power Rule

Inverse

Change of Base

Change of Base & Practical Evaluation

Transformations & Asymptotes

Vertical & Horizontal Shifts

Scale Changes

Worked Examples

Example 1: Simplify

Example 2: Evaluate without Calculator

Example 3: Find Domain & Asymptote

Example 4: Expand and Condense

Practice Problems

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Key Takeaways

Log Scales & Applications

Decibels

pH Scale

Practice Bank

Bank 1: Expand Products

Bank 2: Condense Sums

Bank 3: Change of Base

Bank 4: Domain Analysis

Bank 5: Transformations

Bank 6: Mixed Bases

Bank 7: Identity Pitfalls

Bank 8: Composition with Exp

Bank 9: Real-World Scales

Bank 10: Log Inequalities

Bank 11: Parameter Effects

Bank 12: Numeric Stability

Bank 13: Piecewise Domains

Bank 14: Modeling with Logs

Bank 15: Composition & Inverses

Bank 16: Challenge Mix

FAQ (Extended)