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

Solve the positive-argument inequality.
Locate the asymptote.
Map parent points.
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 $log_3(4-2x)$ .

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 $x<2$ and VA $x=2$ .
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 $log_3(4-2x)$ .

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

The negative inside coefficient places the graph to the left.

representation example

VA $log_2(x+4)$ .

Solution $x=-4$

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

transfer example

X-intercept $log_3(x-2)$ .

Solution $x=3$

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. — **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. — **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. — **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 $ln(x-7)$ .

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Practice

Ten concrete questions

Practice 101

Domain $ln(x-7)$ .

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Practice 202

VA $log_2(x+4)$ .

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Practice 303

X-intercept $log_3(x-2)$ .

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Practice 404

Range of $4lnx+8$ .

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Practice 505

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

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Practice 606

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

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Practice 707

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 808

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

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Practice 909

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

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Practice 1010

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.

Analyze log3(4−2x)log_3(4-2x)log3​(4−2x).

Analyze log3(4−2x)log_3(4-2x)log3​(4−2x).

VA log2(x+4)log_2(x+4)log2​(x+4).

X-intercept log3(x−2)log_3(x-2)log3​(x−2).

Original BetterGrades manuscript, rights-separated references.

Analyze $log_3(4-2x)$ .

Analyze $log_3(4-2x)$ .

VA $log_2(x+4)$ .

X-intercept $log_3(x-2)$ .