Teacher Guide: Solving Logarithmic Equations

Solve equations with the unknown inside a logarithm, reject extraneous solutions, and use logarithms to solve exponential equations like 3^x = 20.

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10 questions on Logarithms. Students join with a name, you see everyone's score.

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Learning Objectives

Solve logarithmic equations by rewriting them in exponential form
Solve equations with logarithms on both sides using the one-to-one property
Condense sums and differences of logarithms before solving, and identify and reject extraneous solutions
Solve exponential equations with different bases by taking logarithms, giving exact and rounded answers

Prerequisites

• Definition of a Logarithm (converting between logarithmic and exponential form)
• Product, Quotient and Power Rule for Logarithms
• Change of Base Formula
• Solving Exponential Equations (same-base method)
• Solving linear and quadratic equations by factoring

Discussion Starters

1. Why may you remove the logarithms in $\log_5 A = \log_5 B$ , but not in $\log_5 A + \log_5 B = 2$ ?
2. The equation $\log_2 x + \log_2(x - 2) = 3$ gives the candidates $4$ and $-2$ . Where does the second one come from?
3. Can $\log_3 x = -2$ have a solution even though the right side is negative? What about $\log_3 x = 0$ ?
4. Before using the calculator, how can you tell that the solution of $3^x = 20$ lies between $2$ and $3$ ?

Common Misconceptions

Every number found while solving is a solution

Remediation: Make the domain check a fixed last step. Show both graphs of $\log_2 x + \log_2(x - 2)$ and $\log_2\big(x(x - 2)\big)$ : they agree for $x > 2$ , but only the condensed one exists for $x < 0$ .

A negative value of $x$ is always extraneous

Remediation: Use $\log_2(x + 5) = 1$ , where $x = -3$ is a valid solution. Emphasize that the arguments must be positive, not $x$ itself.

$\frac{\log 20}{\log 3}$ equals $\log \frac{20}{3}$

Remediation: Let students evaluate both on the calculator: $2.73$ versus about $0.82$ . The quotient rule is about the logarithm of a quotient, not a quotient of logarithms.

Differentiation Ideas

For Struggling Students:

• Start with direct equations like $\log_2 x = 5$ and practice only the conversion to exponential form
• Give a fixed checklist: condense, rewrite, solve, check the domain
• Provide a table of powers of 2, 3, 5 and 10 to recognize values like $81 = 3^4$

For On-Level Students:

• Solve equations with logarithms on both sides and with an expression inside the logarithm
• Solve equations that condense to a quadratic and reject the extraneous candidate
• Solve exponential equations like $5^x = 40$ and round the answer to two decimal places

For Advanced Students:

• Solve equations with logarithms of different bases by first applying the change of base formula
• Solve exponential equations with unknowns on both sides, such as $2^{x+1} = 3^x$ , by taking logarithms
• Construct an equation where both candidates are extraneous and explain why it has no solution

Standards Alignment

HSF-LE.A.4 (CCSS.MATH.CONTENT.HSF.LE.A.4)
For exponential models, express as a logarithm the solution to ab^(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology
HSF-BF.B.5 (CCSS.MATH.CONTENT.HSF.BF.B.5)
Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents
HSA-REI.A.2 (CCSS.MATH.CONTENT.HSA.REI.A.2)
Solve simple equations in one variable and give examples showing how extraneous solutions may arise
HSA-REI.D.11 (CCSS.MATH.CONTENT.HSA.REI.D.11)
Explain why the x-coordinates of the points where the graphs of y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x)

Lesson Resources

visualTwo Graphs, One Equation
Students compare the graphs of the original and the condensed equation and see where the extraneous solution comes from
activityAccept or Reject
Cards with an equation and a candidate: students substitute and decide whether every argument stays positive
worksheetLog Equation Practice
Equations from exponential form to condensing with extraneous solutions, with a domain-check column

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A logarithmic equation is an equation in which the unknown appears inside a logarithm, for example

\log_2 x = 3

\log_3(2x + 1) = 2

Two tools solve almost every logarithmic equation:

1. Rewrite in exponential form. A logarithm is an exponent, so

\log_b A = c \iff A = b^c

For example,

\log_2 x = 3

means

x = 2^3 = 8

. In the graph, the line

y = 3

meets the curve

y = \log_2 x

exactly at

x = 8

2. Use the one-to-one property. If two logarithms with the same base are equal, their arguments are equal:

\log_b M = \log_b N \implies M = N

The method:

1.Condense each side into a single logarithm with the product, quotient and power rules.
2.Rewrite in exponential form, or set the arguments equal.
3.Solve the resulting equation (usually linear or quadratic).
4.Check the domain: every argument in the ORIGINAL equation must be positive. A candidate that makes an argument zero or negative is an extraneous solution and must be rejected.

Exponential equations with different bases. When the two sides of an exponential equation cannot be written with the same base, take the logarithm of both sides and use the power rule:

3^x = 20 \implies x \log 3 = \log 20 \implies x = \frac{\log 20}{\log 3} \approx 2.73

Here

\log

is the common logarithm (base 10). The natural logarithm

\ln

gives the same result.

Worked Examples

Solve $\log_2 x = 3$ .

Identify the parts

Base $2$ , argument $x$ , value of the logarithm $3$ → $\log_2 x = 3$

Rewrite in exponential form

$\log_b A = c \iff A = b^c$ → $x = 2^3$

Calculate the power

$2^3 = 2 \cdot 2 \cdot 2$ → $x = 8$

Check

$\log_2 8 = 3$ because $2^3 = 8$ ✓, and the argument $8$ is positive → The solution is valid

Answer: $x = 8$

Common Mistakes

Keeping both candidates: $\log_2 x + \log_2(x - 2) = 3 \implies x = 4$ or $x = -2$

Why it's wrong: After condensing to $\log_2\big(x(x - 2)\big)$ , negative values of $x$ seem allowed, because the product of two negative numbers is positive. But the original equation contains $\log_2 x$ , which is not defined for $x = -2$ .

Correct: Check every candidate in the ORIGINAL equation. Only $x = 4$ works; $x = -2$ is extraneous.

Dropping the logarithms: $\log_2 x + \log_2(x - 2) = 3 \implies x + (x - 2) = 3$

Why it's wrong: You may remove logarithms only when each side is a single logarithm with the same base (one-to-one property). Here the left side is a sum of logarithms and the right side is a number.

Correct: Condense first, $\log_2\big(x(x - 2)\big) = 3$ , then rewrite in exponential form: $x(x - 2) = 2^3 = 8$ .

Swapping base and exponent: $\log_3(2x + 1) = 2 \implies 2x + 1 = 2^3$

Why it's wrong: In $\log_b A = c$ the base $b$ is raised to the power $c$ . The base of the logarithm stays the base of the power.

Correct: $2x + 1 = 3^2 = 9$ , so $x = 4$ . Check: $\log_3 9 = 2$ ✓

Dividing instead of taking logarithms: $3^x = 20 \implies x = \frac{20}{3} \approx 6.67$

Why it's wrong: $3^x$ is a power, not the product $3 \cdot x$ . A quick check shows the error: $3^{6.67}$ is about $1500$ , far more than $20$ .

Correct: $x = \frac{\log 20}{\log 3} \approx 2.73$ . Note that $\frac{\log 20}{\log 3}$ is NOT the same as $\log \frac{20}{3}$ .

Why It Matters

Logarithmic equations answer "how long?" and "how strong?" questions whenever something grows or shrinks by a constant factor:

Finance: How many years until savings at 8% interest double? Solve $1.08^t = 2$ .
Biology: When does a bacteria culture reach a target size that is not a power of 2?
Sound: How intense is a 90 dB lawn mower compared with the quietest sound you can hear?
Chemistry and geology: pH values and earthquake magnitudes are logarithms, so finding the concentration or energy behind them means solving a logarithmic equation.

In the lesson Solving Exponential Equations you could only say that the solution of

2^x = 20

lies between

4

and

5

. With logarithms you get the exact value

x = \frac{\log 20}{\log 2} \approx 4.32

. And checking the domain is a habit you will need again for every equation with roots, fractions or logarithms.

Real World Applications

Doubling Your Savings

Money that grows by a fixed percentage each year follows $A = P \cdot (1 + r)^t$. Solving for the exponent $t$ with logarithms tells you when a savings goal is reached.

Example:

At 6% interest a year, money doubles when $1.06^t = 2$ , so $t = \frac{\ln 2}{\ln 1.06} \approx 11.9$ years.

1Try It Yourself

You put 1000 dollars into a savings account that pays 8% interest a year, so after $t$ years you have $1000 \cdot 1.08^t$ dollars.

After how many years has the money doubled to 2000 dollars?

Step 1: Write the mathematical expression

Solve $1000 \cdot 1.08^t = 2000$ and write the exact value of $t$ as a quotient of two logarithms, in the form ln(a)/ln(b) (do not round yet):

Loudness in Decibels

The sound level in decibels is $L = 10 \log r$, where $r = \frac{I}{I_0}$ compares the intensity of a sound with the quietest sound people can hear.

Example:

Normal conversation is about 60 dB: $10 \log r = 60$ gives $\log r = 6$ , so $r = 10^6$ , one million times the threshold of hearing.

2Try It Yourself

A lawn mower measures 90 dB. Its intensity ratio $r$ satisfies $10 \log r = 90$ .

What is the intensity ratio $r$ ?

Step 1: Write the mathematical expression

Write $r$ as a power of 10:

Growing Bacteria

A colony that doubles every hour follows $N = N_0 \cdot 2^t$. When the target is not a power of 2, logarithms give the exact time.

Example:

A colony of 200 cells reaches 1000 cells when $2^t = 5$ , so $t = \frac{\ln 5}{\ln 2} \approx 2.32$ hours.

3Try It Yourself

A culture starts with 500 bacteria and doubles every hour.

After how many hours are there 6000 bacteria?

Step 1: Write the mathematical expression

Solve $500 \cdot 2^t = 6000$ and write the exact value of $t$ as a quotient of two logarithms, in the form ln(a)/ln(b) (do not round yet):

Key Takeaways

1A logarithmic equation has the unknown inside a logarithm, like $\log_2 x = 3$
2Rewrite in exponential form: $\log_b A = c \iff A = b^c$
3One-to-one property: $\log_b M = \log_b N$ gives $M = N$
4Condense sums and differences of logarithms into a single logarithm before you solve
5Check every candidate in the ORIGINAL equation: all arguments must be positive, otherwise the candidate is an extraneous solution
6For exponential equations with different bases, take logarithms: $3^x = 20 \implies x = \frac{\log 20}{\log 3} \approx 2.73$

Frequently Asked Questions

Why do extraneous solutions appear at all?

Condensing changes the domain.

\log_2 x + \log_2(x - 2)

needs

x > 2

, but

\log_2\big(x(x - 2)\big)

only needs

x(x - 2) > 0

, which is also true for

x < 0

. The condensed equation can therefore have solutions the original one does not have. That is why you always check in the original equation.

Is a negative solution always extraneous?

No. What matters is the sign of each argument, not the sign of

x

. In

\log_2(x + 5) = 1

we get

x + 5 = 2

, so

x = -3

, and the argument

-3 + 5 = 2

is positive. The solution

x = -3

is valid.

Should I use log or ln to solve $3^x = 20$ ?

Either one.

\frac{\log 20}{\log 3}

and

\frac{\ln 20}{\ln 3}

give the same number, about

2.73

. This is exactly the change of base formula: both are equal to

\log_3 20

Can a logarithmic equation have no solution?

Yes. For

\log_2(x - 4) = \log_2(2x - 1)

the arguments give

x - 4 = 2x - 1

, so

x = -3

. But then both arguments are

-7

, which is not allowed. The only candidate is rejected, so the equation has no solution.

Glossary

Logarithmic equation: An equation in which the unknown appears inside a logarithm, such as $\log_3(2x + 1) = 2$
Exponential form: The power form of a logarithm: $\log_b A = c$ is written as $A = b^c$
One-to-one property: If $\log_b M = \log_b N$ , then $M = N$ (for $M, N > 0$ )
Candidate: A value found while solving that still has to be checked in the original equation
Extraneous solution: A candidate that makes an argument of the original equation zero or negative and is therefore rejected
Domain of a logarithm: The values for which $\log_b A$ is defined: the argument must be positive, $A > 0$

Formula Card

Exponential Form

\log_b A = c \iff A = b^c

Valid for $b > 0$, $b \neq 1$. Use it when one side is a single logarithm and the other side is a number.

One-to-One Property

\log_b M = \log_b N \implies M = N

Use it when both sides are single logarithms with the same base.

Domain Check

A > 0

Every argument $A$ of a logarithm in the original equation must be positive. Otherwise the candidate is extraneous.

Logarithm of Both Sides

b^x = c \implies x = \frac{\log c}{\log b} = \frac{\ln c}{\ln b}

Solves exponential equations whose sides cannot be written with the same base, for example $3^x = 20$.

Teacher Guide: Solving Logarithmic Equations

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Doubling Your Savings

Loudness in Decibels

Growing Bacteria

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Exponential Form

One-to-One Property

Domain Check

Logarithm of Both Sides

More in This Topic

Introduction to Logarithms

Definition of a Logarithm

Common Logarithms (Base 10)

Natural Logarithms (ln)