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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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All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

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⁡48\log_4 8 by choosing a convenient new base
  • Use the reciprocal rule log⁡ba=1log⁡ab\log_b a = \dfrac{1}{\log_a b}
Prerequisites
  • • Definition of a Logarithm: log⁡bx=y\log_b x = y means by=xb^y = x
  • • Common Logarithms (Base 10) and Natural Logarithms (ln): log and ln on a calculator
  • • Power Rule for Logarithms: log⁡(by)=ylog⁡b\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⁡7100\log_7 100?
  • 2. Why do you think log⁡50log⁡3\dfrac{\log 50}{\log 3} and ln⁡50ln⁡3\dfrac{\ln 50}{\ln 3} give exactly the same number?
  • 3. Before calculating, between which two whole numbers is log⁡230\log_2 30? How do you know?
  • 4. If log⁡25≈2.32\log_2 5 \approx 2.32, what can you say about log⁡52\log_5 2 without a calculator?
Common Misconceptions

The base goes on top of the fraction

Remediation: Always estimate first: log⁡350\log_3 50 is between 3 and 4 because 27<50<8127 < 50 < 81. An answer of 0.280.28 is then clearly wrong. Mnemonic: the base is the foundation, so it sits at the bottom.

log⁡alog⁡b\dfrac{\log a}{\log b} is the same as log⁡ab\log \dfrac{a}{b}

Remediation: Evaluate both for a=100a = 100, b=10b = 10: log⁡100log⁡10=21=2\dfrac{\log 100}{\log 10} = \dfrac{2}{1} = 2, but log⁡10=1\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 xx.

Differentiation Ideas

For Struggling Students:

  • • Start with values that come out whole, like log⁡232=log⁡32log⁡2=5\log_2 32 = \dfrac{\log 32}{\log 2} = 5, to build trust in the formula
  • • Give a template: log⁡□△=log⁡△log⁡□\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⁡927\log_9 27 and log⁡832\log_8 32 with a convenient base

For Advanced Students:

  • • Prove log⁡ba⋅log⁡ab=1\log_b a \cdot \log_a b = 1 and simplify chains like log⁡23⋅log⁡34⋅log⁡48\log_2 3 \cdot \log_3 4 \cdot \log_4 8
  • • Show that log⁡bnxn=log⁡bx\log_{b^n} x^n = \log_b x with change of base
  • • Explain why the graph of y=log⁡bxy = \log_b x is a vertical stretch of y=ln⁡xy = \ln x by the factor 1ln⁡b\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=log⁡xlog⁡by = \dfrac{\log x}{\log b} and y=ln⁡xln⁡by = \dfrac{\ln x}{\ln b} with a slider for bb: the curves lie exactly on top of each other

  • activityEstimate First, Then Calculate

    Students bracket log⁡bx\log_b x between two whole numbers with powers of bb, 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 ee). The change of base formula lets you evaluate a logarithm of any base with one of these keys:
log⁡bx=log⁡cxlog⁡cb\log_b x = \frac{\log_c x}{\log_c b}
Here x>0x > 0, b>0b > 0, b≠1b \neq 1, and the new base cc can be any positive number other than 11. In practice you choose c=10c = 10 or c=ec = e:
log⁡bx=log⁡xlog⁡b=ln⁡xln⁡b\log_b x = \frac{\log x}{\log b} = \frac{\ln x}{\ln b}
Why it works: let y=log⁡bxy = \log_b x. Then by=xb^y = x. Take the common logarithm of both sides: log⁡(by)=log⁡x\log(b^y) = \log x. The power rule turns this into y⋅log⁡b=log⁡xy \cdot \log b = \log x, so y=log⁡xlog⁡by = \dfrac{\log x}{\log b}.
Example: log⁡350=log⁡50log⁡3≈1.698970.47712≈3.56\log_3 50 = \dfrac{\log 50}{\log 3} \approx \dfrac{1.69897}{0.47712} \approx 3.56 (rounded to 2 decimal places). Check: 33.56≈503^{3.56} \approx 50.
A useful special case: swapping the base and the argument gives the reciprocal,
log⁡ba=1log⁡ab\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⁡350\log_3 50. Round to 2 decimal places.
1

Write the change of base formula with base 10

log⁡350=log⁡50log⁡3\log_3 50 = \dfrac{\log 50}{\log 3}
2

Find both common logarithms on the calculator

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

Divide, then round only at the end

1.69897÷0.47712≈3.560881.69897 \div 0.47712 \approx 3.56088 → ≈3.56\approx 3.56
4

Check with the definition

33=273^3 = 27 and 34=813^4 = 81, so the answer must lie between 3 and 4. Also 33.56≈503^{3.56} \approx 50 → Reasonable

Common Mistakes

Writing the fraction upside down: log⁡350=log⁡3log⁡50≈0.28\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: 33=273^3 = 27 and 34=813^4 = 81, so log⁡350\log_3 50 must be between 3 and 4, not below 1.

Correct: Argument on top, base on the bottom: log⁡350=log⁡50log⁡3≈3.56\log_3 50 = \dfrac{\log 50}{\log 3} \approx 3.56

Turning the quotient of logs into the log of a quotient: log⁡50log⁡3=log⁡503≈1.22\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⁡503\log 50 - \log 3 = \log \dfrac{50}{3}. It is about a difference of logs, not a division of logs.

Correct: log⁡50log⁡3\dfrac{\log 50}{\log 3} is a plain division of two numbers: 1.69897÷0.47712≈3.561.69897 \div 0.47712 \approx 3.56

Cancelling the word log: log⁡50log⁡3=503≈16.67\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\log 50 and log⁡3\log 3 first, then divide: ≈3.56\approx 3.56

Rounding too early: 1.700.48≈3.54\dfrac{1.70}{0.48} \approx 3.54

Why it's wrong: Rounding log⁡50\log 50 and log⁡3\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: 3.563.56

Why It Matters

The change of base formula is what makes logarithms usable in practice:
  • Any base on any calculator: log⁡2\log_2, log⁡3\log_3 or log⁡1.05\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⁡1.052=log⁡2log⁡1.05≈14.21\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⁡21000000≈19.93\log_2 1000000 \approx 19.93, so 20 halvings.
  • Biology: a population that triples every hour needs log⁡3500≈5.66\log_3 500 \approx 5.66 hours to grow 500 times bigger.
  • Solving equations: every exponential equation like 2x=502^x = 50 has the answer x=log⁡250x = \log_2 50, which you evaluate with this formula.
It also explains why log⁡\log and ln⁡\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 1.031.03, so the money doubles when 1.03t=21.03^t = 2, that is after t=log⁡1.032t = \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⁡1.032\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⁡21024=10\log_2 1024 = 10 halvings, because 210=10242^{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⁡21000000\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 32=93^2 = 9 times bigger, after 4 hours 34=813^4 = 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⁡3500\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⁡bx=log⁡cxlog⁡cb\log_b x = \dfrac{\log_c x}{\log_c b} for any new base c>0c > 0, c≠1c \neq 1
  • 2On a calculator use c=10c = 10 or c=ec = e: log⁡bx=log⁡xlog⁡b=ln⁡xln⁡b\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 by=xb^y = x gives ylog⁡b=log⁡xy \log b = \log x (power rule), so y=log⁡xlog⁡by = \dfrac{\log x}{\log b}
  • 5Swapping base and argument gives the reciprocal: log⁡ba=1log⁡ab\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. log⁡50log⁡3\dfrac{\log 50}{\log 3} and ln⁡50ln⁡3\dfrac{\ln 50}{\ln 3} are both ≈3.56\approx 3.56. Just use the same kind of logarithm on top and on the bottom.

My calculator has a log⁡b\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⁡48=32\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⁡350≈3.56\log_3 50 \approx 3.56 or 3.56093.5609). Always round only once, at the very end.

Why is log⁡ba⋅log⁡ab=1\log_b a \cdot \log_a b = 1?

With change of base, log⁡ba=log⁡alog⁡b\log_b a = \dfrac{\log a}{\log b} and log⁡ab=log⁡blog⁡a\log_a b = \dfrac{\log b}{\log a}. Multiplying the two fractions cancels everything and leaves 11. So the two logarithms are reciprocals.

Glossary

Change of base formula
The rule log⁡bx=log⁡cxlog⁡cb\log_b x = \dfrac{\log_c x}{\log_c b} that rewrites a logarithm with a new base cc
Base
The number bb in log⁡bx\log_b x, the number that is raised to a power; it must be positive and not 11
Argument
The number xx in log⁡bx\log_b x that you take the logarithm of; it must be positive
Common logarithm
The logarithm with base 10, written log⁡x\log x (the log key)
Natural logarithm
The logarithm with base e≈2.718e \approx 2.718, written ln⁡x\ln x (the ln key)
Reciprocal
The number 1a\dfrac{1}{a} for a number a≠0a \neq 0; log⁡ba\log_b a and log⁡ab\log_a b are reciprocals

Formula Card

Change of base (any base)

log⁡bx=log⁡cxlog⁡cb\log_b x = \dfrac{\log_c x}{\log_c b}

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

With common logarithms

log⁡bx=log⁡xlog⁡b\log_b x = \dfrac{\log x}{\log b}

Use the log key

With natural logarithms

log⁡bx=ln⁡xln⁡b\log_b x = \dfrac{\ln x}{\ln b}

Use the ln key, same result

Reciprocal rule

log⁡ba=1log⁡ab\log_b a = \dfrac{1}{\log_a b}

Swapping base and argument flips the value

More in This Topic

Introduction to Logarithms

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Definition of a Logarithm

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Common Logarithms (Base 10)

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Natural Logarithms (ln)

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