Teacher Guide: Change of Base Formula

Use the change of base formula to evaluate a logarithm of any base with the log or ln key of your calculator.

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

For Teachers

Learning Objectives

State the change of base formula and the conditions on the bases
Derive the formula from the definition of a logarithm and the power rule
Evaluate logarithms of any base with the log or ln key, rounded to a given number of decimal places
Find exact values such as $\log_4 8$ by choosing a convenient new base
Use the reciprocal rule $\log_b a = \dfrac{1}{\log_a b}$

Prerequisites

• Definition of a Logarithm: $\log_b x = y$ means $b^y = x$
• Common Logarithms (Base 10) and Natural Logarithms (ln): log and ln on a calculator
• Power Rule for Logarithms: $\log(b^y) = y \log b$
• Rounding to a given number of decimal places

Discussion Starters

1. Your calculator has only log and ln. How could you possibly find $\log_7 100$ ?
2. Why do you think $\dfrac{\log 50}{\log 3}$ and $\dfrac{\ln 50}{\ln 3}$ give exactly the same number?
3. Before calculating, between which two whole numbers is $\log_2 30$ ? How do you know?
4. If $\log_2 5 \approx 2.32$ , what can you say about $\log_5 2$ without a calculator?

Common Misconceptions

The base goes on top of the fraction

Remediation: Always estimate first: $\log_3 50$ is between 3 and 4 because $27 < 50 < 81$ . An answer of $0.28$ is then clearly wrong. Mnemonic: the base is the foundation, so it sits at the bottom.

$\dfrac{\log a}{\log b}$ is the same as $\log \dfrac{a}{b}$

Remediation: Evaluate both for $a = 100$ , $b = 10$ : $\dfrac{\log 100}{\log 10} = \dfrac{2}{1} = 2$ , but $\log 10 = 1$ . Then compare with the quotient rule, which is about subtraction.

Using ln gives a different answer than log

Remediation: Let half the class use log and half use ln for the same problem and compare. The individual logarithms differ, the quotient does not. The graph with both curves shows this for every $x$ .

Differentiation Ideas

For Struggling Students:

• Start with values that come out whole, like $\log_2 32 = \dfrac{\log 32}{\log 2} = 5$ , to build trust in the formula
• Give a template: $\log_{\square} \triangle = \dfrac{\log \triangle}{\log \square}$
• Practice typing the whole quotient into the calculator in one go

For On-Level Students:

• Evaluate logarithms with bases 2, 3, 5, 7 and 1.05 to 2 and 3 decimal places
• Use estimation with powers to check every answer
• Find exact values like $\log_9 27$ and $\log_8 32$ with a convenient base

For Advanced Students:

• Prove $\log_b a \cdot \log_a b = 1$ and simplify chains like $\log_2 3 \cdot \log_3 4 \cdot \log_4 8$
• Show that $\log_{b^n} x^n = \log_b x$ with change of base
• Explain why the graph of $y = \log_b x$ is a vertical stretch of $y = \ln x$ by the factor $\dfrac{1}{\ln b}$

Standards Alignment

HSF-LE.A.4 (CCSS.MATH.CONTENT.HSF.LE.A.4)
Express the solution to an exponential equation as a logarithm with base 2, 10 or e and 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 it to solve problems
HSN-Q.A.3 (CCSS.MATH.CONTENT.HSN.Q.A.3)
Choose a level of accuracy appropriate to limitations on measurement when reporting quantities

Lesson Resources

visualTwo Identical Curves
Graph $y = \dfrac{\log x}{\log b}$ and $y = \dfrac{\ln x}{\ln b}$ with a slider for $b$ : the curves lie exactly on top of each other
activityEstimate First, Then Calculate
Students bracket $\log_b x$ between two whole numbers with powers of $b$ , then check with the calculator
worksheetDoubling Times and Halvings
Real-world problems with interest, binary search and growth, each rounded to a stated number of decimal places

Lesson Content

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

Definition

Most calculators only have two logarithm keys: log (base 10) and ln (base

e

). The change of base formula lets you evaluate a logarithm of any base with one of these keys:

\log_b x = \frac{\log_c x}{\log_c b}

Here

x > 0

b > 0

b \neq 1

, and the new base

c

can be any positive number other than

1

. In practice you choose

c = 10

c = e

\log_b x = \frac{\log x}{\log b} = \frac{\ln x}{\ln b}

Why it works: let

y = \log_b x

. Then

b^y = x

. Take the common logarithm of both sides:

\log(b^y) = \log x

. The power rule turns this into

y \cdot \log b = \log x

, so

y = \dfrac{\log x}{\log b}

Example:

\log_3 50 = \dfrac{\log 50}{\log 3} \approx \dfrac{1.69897}{0.47712} \approx 3.56

(rounded to 2 decimal places). Check:

3^{3.56} \approx 50

A useful special case: swapping the base and the argument gives the reciprocal,

\log_b a = \frac{1}{\log_a b}

Rounding: do the whole division on the calculator and round only the final answer, to the number of decimal places the question asks for.

Worked Examples

Evaluate $\log_3 50$ . Round to 2 decimal places.

Write the change of base formula with base 10

$\log_3 50 = \dfrac{\log 50}{\log 3}$

Find both common logarithms on the calculator

$\log 50 \approx 1.69897$ and $\log 3 \approx 0.47712$ → Keep all the digits for now

Divide, then round only at the end

$1.69897 \div 0.47712 \approx 3.56088$ → $\approx 3.56$

Check with the definition

$3^3 = 27$ and $3^4 = 81$ , so the answer must lie between 3 and 4. Also $3^{3.56} \approx 50$ → Reasonable

Answer: $\log_3 50 \approx 3.56$

Common Mistakes

Writing the fraction upside down: $\log_3 50 = \dfrac{\log 3}{\log 50} \approx 0.28$

Why it's wrong: The base goes in the denominator. A quick estimate shows the error: $3^3 = 27$ and $3^4 = 81$ , so $\log_3 50$ must be between 3 and 4, not below 1.

Correct: Argument on top, base on the bottom: $\log_3 50 = \dfrac{\log 50}{\log 3} \approx 3.56$

Turning the quotient of logs into the log of a quotient: $\dfrac{\log 50}{\log 3} = \log\dfrac{50}{3} \approx 1.22$

Why it's wrong: The quotient rule says $\log 50 - \log 3 = \log \dfrac{50}{3}$ . It is about a difference of logs, not a division of logs.

Correct: $\dfrac{\log 50}{\log 3}$ is a plain division of two numbers: $1.69897 \div 0.47712 \approx 3.56$

Cancelling the word log: $\dfrac{\log 50}{\log 3} = \dfrac{50}{3} \approx 16.67$

Why it's wrong: log is a function, not a factor. You cannot cancel it like a common number in a fraction.

Correct: Evaluate $\log 50$ and $\log 3$ first, then divide: $\approx 3.56$

Rounding too early: $\dfrac{1.70}{0.48} \approx 3.54$

Why it's wrong: Rounding $\log 50$ and $\log 3$ before dividing changes the second decimal place of the answer.

Correct: Divide with all the calculator digits (or type log(50)/log(3) in one go) and round only the final result: $3.56$

Why It Matters

The change of base formula is what makes logarithms usable in practice:

Any base on any calculator: $\log_2$ , $\log_3$ or $\log_{1.05}$ become one division with the log or ln key.
Money and growth: the time for savings to double at 5 % per year is $\log_{1.05} 2 = \dfrac{\log 2}{\log 1.05} \approx 14.21$ years.
Computer science: binary search and bits use base 2. Searching 1,000,000 sorted items takes about $\log_2 1000000 \approx 19.93$ , so 20 halvings.
Biology: a population that triples every hour needs $\log_3 500 \approx 5.66$ hours to grow 500 times bigger.
Solving equations: every exponential equation like $2^x = 50$ has the answer $x = \log_2 50$ , which you evaluate with this formula.

It also explains why

\log

and

\ln

are enough: every other logarithm is just one of them divided by a constant.

Real World Applications

How Long Until Your Savings Double?

With yearly compound interest, money is multiplied by the same growth factor every year. The doubling time is a logarithm whose base is that factor.

Example:

At 3 % per year the factor is $1.03$ , so the money doubles when $1.03^t = 2$ , that is after $t = \log_{1.03} 2$ years.

1Try It Yourself

You put 1000 dollars into a savings account that pays 3 % interest per year, compounded once a year.

After how many years has the money doubled?

Step 1: Write the mathematical expression

Write $\log_{1.03} 2$ as a quotient of two common logarithms (type the expression in the form log(a)/log(b), not the rounded value):

Binary Search in Computer Science

Binary search finds an item in a sorted list by halving the list again and again. The number of halvings is a base-2 logarithm, and computers evaluate it with change of base.

Example:

A sorted list of 1024 names needs $\log_2 1024 = 10$ halvings, because $2^{10} = 1024$ .

2Try It Yourself

A database holds 1,000,000 sorted records. Each step of a binary search halves the number of records that are left.

How many halvings are needed to get down to one record?

Step 1: Write the mathematical expression

Write $\log_2 1000000$ as a quotient of two common logarithms (type the expression in the form log(a)/log(b), not the rounded value):

Bacteria That Triple Every Hour

Some bacteria cultures triple every hour. The time to reach a certain size is a base-3 logarithm.

Example:

After 2 hours the culture is $3^2 = 9$ times bigger, after 4 hours $3^4 = 81$ times bigger.

3Try It Yourself

A culture triples every hour. A lab needs it to be 500 times bigger than at the start.

How many hours does that take?

Step 1: Write the mathematical expression

Write $\log_3 500$ as a quotient of two common logarithms (type the expression in the form log(a)/log(b), not the rounded value):

Key Takeaways

1Change of base formula: $\log_b x = \dfrac{\log_c x}{\log_c b}$ for any new base $c > 0$ , $c \neq 1$
2On a calculator use $c = 10$ or $c = e$ : $\log_b x = \dfrac{\log x}{\log b} = \dfrac{\ln x}{\ln b}$ , both give the same answer
3The argument goes on top, the base goes on the bottom
4It works because $b^y = x$ gives $y \log b = \log x$ (power rule), so $y = \dfrac{\log x}{\log b}$
5Swapping base and argument gives the reciprocal: $\log_b a = \dfrac{1}{\log_a b}$
6Divide with all the digits and round only the final answer, to the decimal places asked for

Frequently Asked Questions

Should I use log or ln in the formula?

Either one.

\dfrac{\log 50}{\log 3}

and

\dfrac{\ln 50}{\ln 3}

are both

\approx 3.56

. Just use the same kind of logarithm on top and on the bottom.

My calculator has a $\log_b$ key. Do I still need the formula?

The key gives you the number, but the formula is still needed to find exact answers like

\log_4 8 = \dfrac{3}{2}

, to simplify expressions, to solve equations by hand and on tests where only log and ln are allowed. The calculator key itself uses change of base internally.

How many decimal places should I round to?

As many as the question asks for. If it says nothing, 2 to 4 decimal places is usual (for example

\log_3 50 \approx 3.56

3.5609

). Always round only once, at the very end.

Why is $\log_b a \cdot \log_a b = 1$ ?

With change of base,

\log_b a = \dfrac{\log a}{\log b}

and

\log_a b = \dfrac{\log b}{\log a}

. Multiplying the two fractions cancels everything and leaves

1

. So the two logarithms are reciprocals.

Glossary

Change of base formula: The rule $\log_b x = \dfrac{\log_c x}{\log_c b}$ that rewrites a logarithm with a new base $c$
Base: The number $b$ in $\log_b x$ , the number that is raised to a power; it must be positive and not $1$
Argument: The number $x$ in $\log_b x$ that you take the logarithm of; it must be positive
Common logarithm: The logarithm with base 10, written $\log x$ (the log key)
Natural logarithm: The logarithm with base $e \approx 2.718$ , written $\ln x$ (the ln key)
Reciprocal: The number $\dfrac{1}{a}$ for a number $a \neq 0$ ; $\log_b a$ and $\log_a b$ are reciprocals

Formula Card

Change of base (any base)

\log_b x = \dfrac{\log_c x}{\log_c b}

Any new base $c > 0$, $c \neq 1$

With common logarithms

\log_b x = \dfrac{\log x}{\log b}

Use the log key

With natural logarithms

\log_b x = \dfrac{\ln x}{\ln b}

Use the ln key, same result

Reciprocal rule

\log_b a = \dfrac{1}{\log_a b}

Swapping base and argument flips the value

Teacher Guide: Change of Base Formula

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

How Long Until Your Savings Double?

Binary Search in Computer Science

Bacteria That Triple Every Hour

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Change of base (any base)

With common logarithms

With natural logarithms

Reciprocal rule

More in This Topic

Introduction to Logarithms

Definition of a Logarithm

Common Logarithms (Base 10)

Natural Logarithms (ln)