BetterGrades Precalculus · Unit 1 · Lesson

Exponential and logarithmic algebra repair

Apply exponent and logarithm laws, convert between inverse forms, and solve basic equations with valid domains.

Opening

Start with the situation

Exponent laws describe repeated multiplication, while logarithms reverse exponentiation.

These algebra choices are the load-bearing steps beneath later work with functions. A restriction lost here can turn into a false intercept, a missing asymptote, or an invalid model several lessons later.

Before you begin

Prerequisite check

Use signed arithmetic accurately.
Show algebraic steps in a checkable order.
State restrictions before simplifying.

Core explanation

Explanation

Apply product, quotient, and power laws only to multiplicative structure; convert $log_b(y)=x$ and $b^x=y$ without changing the base-exponent-output roles.

Real logarithm arguments must be positive, and bases must be positive and unequal to $1$ .

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through exponent-law derivation, exponential-log inverse pair, or another equivalent representation.

Conceptual reading

What the idea is really doing

Precalculus is unforgiving about hidden algebra errors. The useful habit is to separate reversible algebra from steps that can create candidates, and to keep domain restrictions beside the work instead of trying to remember them at the end.

This lesson narrows that lens to one goal: apply exponent and logarithm laws, convert between inverse forms, and solve basic equations with valid domains. 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

Apply product.
Quotient.
Power laws only to multiplicative structure; convert $log_b(y)=x$ .
$B^x=y$ without changing the base-exponent-output roles.

Verification: Re-read the original statement, not only the simplified line. Confirm every restriction, substitute each candidate, and describe what the result means before moving on.

Foundation walkthrough

Plan before calculating

Problem

Evaluate $log_3(\frac{1}{27})$

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Apply product, quotient, and power laws only to multiplicative structure; convert $log_b(y)=x$ and $b^x=y$ without changing the base-exponent-output roles.
Conclusion: $-3,$ because $3^(-3)=\frac{1}{27}$ .
Why the check works: A logarithm reports an exponent.

Worked examples

See the idea in three forms

foundation example

Evaluate $log_3(\frac{1}{27})$

Solution $-3,$ because $3^(-3)=\frac{1}{27}$ .

A logarithm reports an exponent.

representation example

Evaluate $log_5(125)$

Solution $3$

This example expresses exponential and logarithmic algebra repair in a second form.

transfer example

Expand $log_b(xy^2)$

Solutionlog_b $x+2log_b y$ .

Real logarithm arguments must be positive, and bases must be positive and unequal to $1$ .

Exponent-law derivation. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: A logarithm reports an exponent. — **Anchor figure · Exponent-law derivation**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: A logarithm reports an exponent.

Exponential-log inverse pair. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for exponential and logarithmic algebra repair. — **Mechanism figure · Exponential-log inverse pair**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for exponential and logarithmic algebra repair.

False sum-law counterexamples. Compare the valid path with the tempting shortcut. The figure shows why applying a product law to a sum, such as log(a+b)=log a+log b leads to a false conclusion. — **Comparison and error figure · False sum-law counterexamples**
Compare the valid path with the tempting shortcut. The figure shows why applying a product law to a sum, such as $log(a+b)=log a+log b$ leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is applying a product law to a sum, such as $log(a+b)=log a+log b$ .

Check yourself

Simplify $\frac{x^5}{x}^2$

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Practice

Ten concrete questions

Practice 101

Simplify $\frac{x^5}{x}^2$

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Practice 202

Evaluate $log_5(125)$

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Practice 303

Expand $log_b(xy^2)$

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Practice 404

Solve $3^x=81$

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Practice 505

Explain why this conclusion is valid: $-3,$ because $3^(-3)=\frac{1}{27}$ . Use the foundation problem as evidence: Evaluate $log_3(\frac{1}{27})$ .

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Practice 606

Solve the representation example, then name the feature of exponential and logarithmic algebra repair that it illustrates: Evaluate $log_5(125)$

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Practice 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is applying a product law to a sum, such as $log(a+b)=log a+log b$ .

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Practice 808

Connect two representations for this example: Evaluate $log_3(\frac{1}{27})$ . 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: Expand $log_b(xy^2)$ . 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 exponential and logarithmic algebra repair, then apply it to one worked example from this lesson.

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Lesson close

Connect forward

The next lesson, Function notation and graph reading repair, uses this result as part of a larger structure.

Source record

Original BetterGrades manuscript, rights-separated references.

Stitz and Zeager, Precalculus Chapter 0
BetterGrades Algebra course
Redden, Advanced Algebra

No long source passage is reproduced.

Evaluatelog3(127)log_3(\frac{1}{27})log3​(271​)

Evaluatelog3(127)log_3(\frac{1}{27})log3​(271​)

Evaluatelog5(125)log_5(125)log5​(125)

Expandlogb(xy2)log_b(xy^2)logb​(xy2)

Original BetterGrades manuscript, rights-separated references.

Evaluate $log_3(\frac{1}{27})$

Evaluate $log_3(\frac{1}{27})$

Evaluate $log_5(125)$

Expand $log_b(xy^2)$