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BetterGrades Precalculus · Unit 7 · Lesson

Logarithmic graphs and transformations

Analyze transformed logarithmic functions, including domain, range, vertical asymptote, intercepts, and base-dependent direction.

Opening

Start with the situation

A transformed logarithm has a vertical input boundary where its argument equals zero.

Multiplicative models describe repeated percentage change, while logarithms recover the time or exponent hidden inside that process. Together they support growth, decay, finance, regression, and bounded models.

Before you begin

Prerequisite check

  • Use exponent laws.
  • Interpret function parameters.
  • Distinguish exact and approximate values.
Core explanation

Explanation

Solve the positive-argument inequality, locate the asymptote, map parent points, and find intercepts by exponential conversion.

A negative inside coefficient places the domain on the opposite side of the asymptote.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through log parent features, transformed boundary, or another equivalent representation.

Conceptual reading

What the idea is really doing

Exponential change multiplies over equal input steps, while logarithms answer the inverse question: what exponent produces a given output? Parameters must be interpreted as an initial value, a multiplier, a rate, or a long-run bound—not as decoration.

This lesson narrows that lens to one goal: analyze transformed logarithmic functions, including domain, range, vertical asymptote, intercepts, and base-dependent direction. The point is not to memorize an isolated trick; it is to know what evidence makes the conclusion valid and how a second representation can check it.

Reusable method

A reliable route through the problem

  1. Solve the positive-argument inequality.
  2. Locate the asymptote.
  3. Map parent points.
  4. Find intercepts by exponential conversion.

Verification: Test the model at input zero and one step later, confirm the multiplier or inverse relationship, and state whether the domain and long-run behavior make sense in context.

Foundation walkthrough

Plan before calculating

Problem

Analyze log3(4−2x)log_3(4-2x).

Plan
Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Solve the positive-argument inequality, locate the asymptote, map parent points, and find intercepts by exponential conversion.
Conclusion
Domain x<2x<2 and VA x=2x=2.
Why the check works
The negative inside coefficient places the graph to the left.
Worked examples

See the idea in three forms

foundation example

Analyze log3(4−2x)log_3(4-2x).

SolutionDomain x<2x<2 and VA x=2x=2.

The negative inside coefficient places the graph to the left.

representation example

VA log2(x+4)log_2(x+4).

Solutionx=−4x=-4

This example expresses logarithmic graphs and transformations in a second form.

transfer example

X-intercept log3(x−2)log_3(x-2).

Solutionx=3x=3

A negative inside coefficient places the domain on the opposite side of the asymptote.

Log parent features. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The negative inside coefficient places the graph to the left.
Read this graph as text

Logarithmic graphs and transformations · Log parent features. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The negative inside coefficient places the graph to the left. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.

Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.

Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Analyze transformed logarithmic functions, including domain, range, vertical asymptote, intercepts, and base-dependent direction.

Anchor figure · Log parent features

Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The negative inside coefficient places the graph to the left.

Transformed boundary. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for logarithmic graphs and transformations.
Read this graph as text

Logarithmic graphs and transformations · Transformed boundary. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for logarithmic graphs and transformations. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.

Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.

Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Analyze transformed logarithmic functions, including domain, range, vertical asymptote, intercepts, and base-dependent direction.

Mechanism figure · Transformed boundary

Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for logarithmic graphs and transformations.

Base comparison. Compare the valid path with the tempting shortcut. The figure shows why choosing the wrong side of the domain boundary leads to a false conclusion.
Read this graph as text

Logarithmic graphs and transformations · Base comparison. Compare the valid path with the tempting shortcut. The figure shows why choosing the wrong side of the domain boundary leads to a false conclusion. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.

Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.

Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Analyze transformed logarithmic functions, including domain, range, vertical asymptote, intercepts, and base-dependent direction.

Comparison and error figure · Base comparison

Compare the valid path with the tempting shortcut. The figure shows why choosing the wrong side of the domain boundary leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is choosing the wrong side of the domain boundary.

Check yourself

Domain ln(x−7)ln(x-7).

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Practice

10 concrete questions

Practice 1 · retrieval · foundational01

Domain ln(x−7)ln(x-7).

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Practice 2 · procedural · developing02

VA log2(x+4)log_2(x+4).

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Practice 3 · procedural · developing03

X-intercept log3(x−2)log_3(x-2).

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Practice 4 · conceptual · transfer04

Range of 4lnx+84lnx+8.

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Practice 5 · explanation · developing05

Explain why this conclusion is valid: Domain x<2x<2 and VA x=2x=2. Use the foundation problem as evidence: Analyze log3(4−2x)log_3(4-2x).

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Practice 6 · procedural · developing06

Solve the representation example, then name the feature of logarithmic graphs and transformations that it illustrates: VAlog2(x+4)log_2(x+4)

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Practice 7 · error analysis · transfer07

Correct this reasoning and identify the first unsafe assumption: A frequent error is choosing the wrong side of the domain boundary.

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Practice 8 · graphical · transfer08

Connect two representations for this example: Analyze log3(4−2x)log_3(4-2x). Describe what a graph, table, mapping, or algebraic form would have to show.

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Practice 9 · transfer · transfer09

Create a nearby example by changing one number or condition in this prompt: X-intercept log3(x−2)log_3(x-2). Predict the effect, solve your new example, and compare it with the original.

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Practice 10 · verification · transfer10

Write a short verification checklist for logarithmic graphs and transformations, then apply it to one worked example from this lesson.

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Lesson close

Connect forward

The next lesson, Logarithm laws, uses this result as part of a larger structure.

Source record

Original BetterGrades manuscript, rights-separated references.

  • Yoshiwara, Modeling, Functions, and Graphs
  • Lippman and Rasmussen, Precalculus Volume 1
  • Stitz and Zeager, Precalculus

No long source passage is reproduced.