Approximate logarithms in any base with technology

Ontario Grade 12 Mathematics

Exponential and Logarithmic Functions

Using technology to find and check logarithmic values

A logarithm answers an exponent question. For example, asking for

\log_2 8

means asking, “What exponent on

2

gives

8

?” Since

2^3=8

, the answer is

3

. Some logarithms are not easy to work out mentally. Technology can approximate them, including logarithms with bases that do not have their own calculator key. In this lesson, an approximation is a decimal value close to the exact value.

What you will learn

Explain a logarithm as an exponent.
Approximate a logarithm when its base is not a calculator’s common log key.
Use technology and the change-of-base relationship to find a logarithm in any valid base.
Check whether a calculated approximation is reasonable.

1. Prerequisite bridge: a logarithm is an exponent

A power has a base and an exponent. In

5^2=25

, the base is

5

and the exponent is

2

. A logarithm reverses this question: it tells you which exponent produces a given number.

The notation

\log_b a

means “the exponent on b that gives

a

.” Here,

b

is the base and

a

is the number being reached. For instance,

\log_3 81=4

because

3^4=81

.

For a real logarithm, the base must be positive and not equal to

1

, and the number being logged must be positive. These conditions ensure that the logarithm represents an exponent question that can be answered with a real number.

\log_b a=x \iff b^x=a

The base of a logarithm becomes the base of the related power.
The logarithm’s value is the exponent.
The number inside the logarithm must be positive.

2. Plain language and calculator notation

A calculator usually has keys for the common logarithm, written

\log x

, and the natural logarithm, written

\ln x

. The common logarithm uses base

10

. The natural logarithm uses base

e

, a number approximately equal to

2.718

. Neither key directly covers every possible base.

Technology can still find a logarithm in any valid base. Many graphing calculators and apps let you enter a base directly. On other calculators, use the change-of-base relationship. It rewrites a logarithm in terms of logarithms that the calculator can evaluate.

When using the change-of-base relationship, use the same type of calculator key in the top and bottom: both common logarithms or both natural logarithms. The ratio gives the same result either way. Keep the full ratio in the calculator before rounding.

A useful estimate comes from nearby powers. If the base is

7

and the number is

20

, then

7^1=7

and

7^2=49

. Therefore,

\log_7 20

must be between

1

and

2

. Technology gives a more precise decimal within that interval.

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

The $\log$ key means base $10$ ; the $\ln$ key means base $e$ .
A calculator that accepts an entered base can evaluate the logarithm directly.
The change-of-base relationship works with either $\log$ or $\ln$ .

3. Representing and interpreting a calculator result

The relationship between a logarithm and a power gives three ways to think about the same value. In words,

\log_7 20

asks which exponent on

7

gives

20

. In a power statement, the unknown exponent is the number

x

that makes

7^x=20

. On a calculator, the change-of-base expression gives a decimal approximation for that exponent.

A table can help you decide whether a result is sensible before you accept it. Compare powers of the base with the number inside the logarithm. If the number lies between two consecutive powers, the logarithm lies between their exponents.

The table uses

7

and

20

as an illustration. Since

7

is greater than

1

, its powers increase as the exponent increases. The number

20

lies between

7

and

49

, so the logarithm is between

1

and

2

. This estimate does not replace the calculator; it checks the size of the answer.

7^1<20<7^2 \Rightarrow 1<\log_7 20<2

Use nearby powers to predict the range of a logarithm.
A decimal approximation should agree with that range.
Rounding changes the displayed value slightly, so keep enough digits for the task.

4. Using technology carefully

For a logarithm with an unusual base, first confirm that both the base and the number inside the logarithm meet the stated conditions. Then choose a calculator method: enter the base directly if the technology supports it, or enter the change-of-base ratio using

\log

or

\ln

.

Use parentheses so the calculator divides the logarithm of the number by the logarithm of the base. Without clear grouping, some calculators may perform operations in an unintended order. Check the display before rounding, and state the result as an approximation rather than an exact value.

Finally, test the approximation in the original power question. If the result is

x

, evaluate the base raised to

x

. The result should be close to the number inside the original logarithm. A small difference is expected because the displayed exponent may be rounded.

This process applies to any permitted base, not just the base in the example. The technology supplies the decimal approximation; the meaning of the logarithm and the power check help you interpret it.

b^{\log_b a}\approx a

Group the numerator and denominator clearly.
Do not round intermediate values too early.
Check the result by raising the base to the approximate exponent.

A range check for $\log_7 20$

Exponent	Power of $7$	Comparison with $20$
$1$	$7^1=7$	Below $20$
?	$7^?=20$	Exponent is between $1$ and $2$
$2$	$7^2=49$	Above $20$

Worked example

Approximate a logarithm with an unusual base

Use a calculator to approximate

\log_7 20

to four decimal places. Check that the result is reasonable.

Predict the range
Compare $20$ with powers of the base. Since $7^1=7$ and $7^2=49$ , the exponent that produces $20$ must be between $1$ and $2$ . This gives a range to check against the calculator.
$7<20<49$
Rewrite for the calculator
Use the change-of-base relationship with the natural logarithm key. The base and the number inside the logarithm are both positive, and the base is not $1$ , so the logarithm is valid.
$\log_7 20=\frac{\ln 20}{\ln 7}$
Evaluate and round
Enter the entire quotient into the calculator. Keep the displayed digits until the final step, then round to four decimal places.
$\frac{\ln 20}{\ln 7}\approx 1.5395$
Check with a power
Raise $7$ to the rounded exponent. The result is close to $20$ , as expected. A slight difference is normal because the exponent was rounded.
$7^{1.5395}\approx 20$

Answer:

\log_7 20\approx 1.5395

.

Check: The approximation is between

1

and

2

, matching the comparison

7^1<20<7^2

. Raising

7

to the rounded exponent gives a value close to

20

.

Common mistakes and how to avoid them

Using the common-log key alone for a logarithm with a different base.

Correction: Either use a calculator that accepts the base directly or enter the change-of-base ratio.

Reversing the numerator and denominator in the change-of-base ratio.

Correction: The logarithm of the number being logged goes on top, and the logarithm of the base goes on the bottom.

Rounding each logarithm before dividing.

Correction: Keep the full calculator values in the ratio and round only the final answer.

Accepting a decimal without checking its size.

Correction: Compare the number being logged with nearby powers of the base, then check the decimal by raising the base to that exponent.

Trying to find a real logarithm of zero or a negative number.

Correction: The number inside a real logarithm must be positive.

Lesson summary

A logarithm gives the exponent needed to produce a number from a base.
For a base not provided as a calculator key, use direct base entry or the change-of-base relationship.
Use nearby powers to estimate a sensible range.
Round only after evaluating and check the result with a power.

Check your understanding

Question 1

Which calculator entry using natural logarithms gives

\log_5 12

?

$\frac{\ln 5}{\ln 12}$
$\frac{\ln 12}{\ln 5}$
$\ln 12-\ln 5$
$\frac{\ln 12}{5}$

Show answer and explanation

\frac{\ln 12}{\ln 5}

The number inside the logarithm goes in the numerator, and the base goes in the denominator.

Question 2

Without calculating a decimal, which interval contains

\log_4 10

?

Between $0$ and $1$
Between $1$ and $2$
Between $2$ and $3$
Between $3$ and $4$

Show answer and explanation

Between

1

and

2

Since

4^1=4

and

4^2=16

, the number

10

is between those powers. Its exponent is therefore between

1

and

2

.

Question 3

A calculator gives

\log_3 15\approx 2.4650

. Which check best matches the meaning of this result?

Check whether $3^{2.4650}$ is close to $15$ .
Check whether $15^{2.4650}$ is close to $3$ .
Check whether $3+2.4650$ is close to $15$ .
Check whether $\frac{15}{2.4650}$ is close to $3$ .

Show answer and explanation

Check whether

3^{2.4650}

is close to

15

.

The logarithm is the exponent on

3

that produces

15

, so raise

3

to the approximate exponent.

Key terms

Logarithm: The exponent that tells how many times a base must be used as a factor to produce a given positive number.
Base: The number being raised to an exponent in a power or serving as the base of a logarithm.
Common logarithm: A logarithm with base $10$ , usually written $\log x$ .
Natural logarithm: A logarithm with base $e$ , usually written $\ln x$ .
Approximation: A value close to an exact value, often written as a rounded decimal.
Change of base: A relationship that rewrites a logarithm in one base as a ratio of logarithms in a calculator-friendly base.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation A1.2. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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