MATH 130 · Section 4.3

Logarithmic Functions

Inverse relationships, graphs, equations, and applications

55 minutes · Offline capable

Today’s Learning Roadmap

  1. 1

    convert between logarithmic and exponential forms

  2. 2

    evaluate and simplify logarithmic expressions

  3. 3

    graph logarithmic functions and determine their domains

  4. 4

    solve logarithmic and exponential equations using inverse properties

Activate · 0.5 minSource: 2

Predict: convert between logarithmic and exponential forms

Predict first

If $2^5=32$, what question is $\log_2(32)$ answering?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

convert between logarithmic and exponential forms

Formula focus
  • $\log_b(x)=y$ means exactly $b^y=x$.
  • The logarithm is the exponent.

Explain · 2.5 minSource: 3, 4

Model the Skill: convert between logarithmic and exponential forms

1

$\log_4(64)=3$ means $4^3=64$.

2

$4^{-3}=1/64$ means $\log_4(1/64)=-3$.

3

The base remains 4 and the logarithm records the exponent.

Model · 3.0 minSource: 5

You Try: convert between logarithmic and exponential forms

Student practice

Rewrite $5^3=125$, $3^{-2}=1/9$, and $\ln(7)=x$ in the opposite form.

Practice · 3.0 minSource: 6

Check and Diagnose: convert between logarithmic and exponential forms

Error analysis
  • Keep the base unchanged.
  • The log argument becomes the exponential result.
Self-check

What evidence confirms the setup, sign, unit, or magnitude?

Feedback · 2.0 minSource: 7

Pacing checkpoint

Make It Stick: convert between logarithmic and exponential forms

Complete the sentence: a logarithm tells us ____.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: 8

Predict: evaluate and simplify logarithmic expressions

Predict first

Without a calculator, evaluate $\log_2(32)$ and $\ln(1)$.

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

evaluate and simplify logarithmic expressions

Formula focus
  • Common log has base 10; natural log has base e.
  • $\log_b(1)=0$, $\log_b(b)=1$, and $\log_b(b^x)=x$.

Explain · 2.5 minSource: 9, 10

Model the Skill: evaluate and simplify logarithmic expressions

1

$\log(250)\approx2.398$, and $10^{2.398}\approx250$.

2

$\ln(40)\approx3.689$, and $e^{3.689}\approx40$.

3

The inverse checks confirm both calculator entries.

Model · 3.0 minSource: 11

You Try: evaluate and simplify logarithmic expressions

Student practice

Simplify $\log_7(1)$, $\ln(e^{-5/2})$, and $e^{\ln 7}$.

Practice · 3.0 minSource: 12, 13

Check and Diagnose: evaluate and simplify logarithmic expressions

Error analysis
  • A logarithm of a number between 0 and 1 is negative when the base exceeds 1.
  • Logs undo exponents only when bases match.
Self-check

What evidence confirms the setup, sign, unit, or magnitude?

Feedback · 2.0 minSource: 14

Pacing checkpoint

Make It Stick: evaluate and simplify logarithmic expressions

Name two inverse identities involving logarithms and exponents.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: 15

Predict: graph logarithmic functions and determine their domains

Predict first

What feature of an exponential graph becomes the vertical asymptote of its inverse?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

graph logarithmic functions and determine their domains

Concept
  • Logarithmic graphs reflect exponential graphs across $y=x$.
  • For $f(x)=\log_b(g(x))$, require $g(x)>0$.

Explain · 2.5 minSource: 16

Model the Skill: graph logarithmic functions and determine their domains

Annotated diagram
  • Transform $y=\log_2(x)$ into $f(x)=\log_2(x+1)+2$.
  • Shift the parent graph left 1 and up 2.
  • Domain: $(-1,\infty)$; vertical asymptote: $x=-1$.
  • Intercepts: $(0,2)$ and $(-3/4,0)$.
Graph of f of x equals log base 2 of x plus 1, plus 2, with vertical asymptote x equals negative 1 and labeled intercepts
Graph of f of x equals log base 2 of x plus 1, plus 2, with vertical asymptote x equals negative 1 and labeled intercepts

Model · 3.0 minSource: 17

You Try: graph logarithmic functions and determine their domains

Student practice

Find the domain and vertical asymptote of $\log(3x-6)$.

Practice · 3.0 minSource: 18

Check and Diagnose: graph logarithmic functions and determine their domains

Error analysis
  • The logarithm argument must be strictly positive, not merely nonnegative.
  • Horizontal shifts move the vertical asymptote.
Self-check

What evidence confirms the setup, sign, unit, or magnitude?

Feedback · 2.0 minSource: 19

Pacing checkpoint

Make It Stick: graph logarithmic functions and determine their domains

Explain how to find a logarithmic domain before graphing.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: 20

Predict: solve logarithmic and exponential equations using inverse properties

Predict first

When should you convert to exponential form, and when should you take a logarithm?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

solve logarithmic and exponential equations using inverse properties

Formula focus
  • Use exponential form for a single logarithm.
  • Use one-to-one properties when equal logs have the same base.
  • Use change of base: $\log_a(x)=\log(x)/\log(a)$.

Explain · 2.5 minSource: 21, 22

Model the Skill: solve logarithmic and exponential equations using inverse properties

1

$x-5=2^4=16$, so $x=21$; the argument $16$ is positive.

2

$\log_4(25)=\ln(25)/\ln(4)\approx2.322$.

3

Substitution and exponentiation verify both results.

Model · 3.0 minSource: 23, 24

You Try: solve logarithmic and exponential equations using inverse properties

Student practice

Solve $3^x=175$ and evaluate $\log_3(50)$.

Practice · 3.0 minSource: 25

Check and Diagnose: solve logarithmic and exponential equations using inverse properties

Error analysis
  • Reject solutions that make a log argument nonpositive.
  • The log of a sum is not the sum of logs.
Self-check

What evidence confirms the setup, sign, unit, or magnitude?

Feedback · 2.0 minSource: 26, 27

Pacing checkpoint

Make It Stick: solve logarithmic and exponential equations using inverse properties

Write a two-step checklist for solving and checking a logarithmic equation.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: 28

Session Summary and Exit Ticket

  1. State the most important idea from today without looking at your notes.
  2. Complete one representative setup and identify the step most likely to cause an error.
  3. Write one question that should be answered before the next assessment.
Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 1.5 minSource: 29, 30, 31