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RF7 · Relate logarithms to exponential statements

Learn to relate logarithms to exponential statements through clear examples and targeted practice.

Alberta Grade 12 Mathematics

Relations and Functions

Math 30-1 study topic RF7

A logarithm answers a question about an exponent. For example, the statement 23=82^3=8 tells us that 2 must be multiplied by itself three times to make 8. The related logarithmic statement, log⁡28=3\log_2 8=3, says that the exponent on 2 is 3. The two statements contain the same information, but they highlight different parts of it. In this lesson, you will practise moving between the two forms and checking that each statement makes sense.

What you will learn

  • Explain how a logarithmic statement and an exponential statement express the same relationship.
  • Rewrite exponential statements as logarithmic statements, and rewrite logarithmic statements as exponential statements.
  • Identify the base, exponent, and result in each form.
  • Check whether a logarithmic statement is defined.

1. Review powers and read the notation

A power has a base and an exponent. In 52=255^2=25, the base is 5 and the exponent is 2. The equation says that 5 multiplied by itself twice is 25. The value on the right is the result of the power.
A logarithm uses the notation log⁡bN\log_b N. The small number bb is the base, and NN is called the argument. The logarithm asks: “To what exponent must I raise bb to get NN?” The answer is the value of the logarithm.
For instance, log⁡525=2\log_5 25=2 asks what exponent on 5 gives 25. Since 52=255^2=25, the answer is 2. The logarithm does not mean that the base is multiplied by the argument.
bx=N  ⟺  log⁡bN=xb^x=N\;\Longleftrightarrow\;\log_b N=x
  • In bx=Nb^x=N, bb is the base, xx is the exponent, and NN is the result.
  • In log⁡bN=x\log_b N=x, bb is the base, NN is the argument, and xx is the answer.

2. The conversion rule and its restrictions

The key rule is that an exponential statement and its matching logarithmic statement are equivalent. Equivalent statements have the same meaning. To convert bx=Nb^x=N into logarithmic form, keep the base bb, put the result NN after the logarithm, and set the logarithm equal to the exponent xx.
To convert in the other direction, read log⁡bN=x\log_b N=x as “the exponent on bb that gives NN is xx.” Then write bx=Nb^x=N. This reverse check is useful because it shows whether the base, exponent, and result have been placed correctly.
For real-number logarithms, the base must be positive and cannot equal 1. The argument must be positive. These restrictions come from the matching power statement: a positive base other than 1 is used, and its real powers have positive results. Thus, a statement such as log⁡3(−9)\log_3(-9) is not defined as a real logarithm. A base of 1 is also not allowed, since powers of 1 cannot produce different positive results.
When the base and result are powers that are easy to evaluate, use exponent facts to find the exact answer. For example, 43=644^3=64 gives log⁡464=3\log_4 64=3. Do not use a calculator approximation when the exact exponent is already clear.
b>0,b≠1,N>0b>0,\quad b\ne1,\quad N>0
  • The base stays the base in both forms.
  • The exponent becomes the value of the logarithm.
  • The result of the power becomes the logarithm's argument.
  • A real logarithm needs b>0b>0, b≠1b\ne1, and N>0N>0.

3. A table and a graph show the same relationship

A short table makes the matching statements visible. For base 2, the powers 202^0, 212^1, 222^2, and 232^3 give 1, 2, 4, and 8. Each row can be read in two ways: as a power statement or as a logarithm statement.
For example, the row 22=42^2=4 also gives log⁡24=2\log_2 4=2. The logarithm records which exponent in the power statement produced the listed result.
The graph of y=2xy=2^x plots each exponent xx against its result yy. The graph of y=log⁡2xy=\log_2 x reverses those roles: the input is a positive result, and the output is the exponent that produces it. For example, the points (3,8)(3,8) and (8,3)(8,3) match the statements 23=82^3=8 and log⁡28=3\log_2 8=3. The coordinates switch order because the two forms answer opposite versions of the same question.
(x,y) on y=2x  ⟺  (y,x) on y=log⁡2x(x,y)\text{ on }y=2^x\;\Longleftrightarrow\;(y,x)\text{ on }y=\log_2 x
  • A power statement gives a result from a chosen exponent.
  • A logarithmic statement gives the exponent for a chosen positive result.
  • The exponent and result exchange roles when the two forms are compared.

4. A reliable conversion routine

When converting a statement, first identify its base. Next, locate the exponent and the result. In logarithmic form, the result must be the argument and the exponent must be the value of the logarithm. After converting, read the new statement back as a power to check it.
If a statement contains a variable, the same rule applies. For example, log⁡3x=4\log_3 x=4 means that 3 raised to the fourth power equals xx. Therefore x=81x=81. The original logarithm also requires x>0x>0, which is satisfied by 81.
A conversion does not by itself solve every equation. It changes how the relationship is written. If a variable is the argument, use the equivalent power statement to find its value, then check that the value is allowed in the original logarithm.
log⁡bN=x  ⟺  bx=N\log_b N=x\;\Longleftrightarrow\;b^x=N
  • Keep the base fixed while changing forms.
  • Check a conversion by rewriting it as a power statement.
  • For a variable argument, check that the resulting value is positive.

Two readings of powers of 2

Exponential statementLogarithmic statement
20=12^0=1log⁡21=0\log_2 1=0
21=22^1=2log⁡22=1\log_2 2=1
22=42^2=4log⁡24=2\log_2 4=2
23=82^3=8log⁡28=3\log_2 8=3

Worked example

Convert an exponential statement

Write 34=813^4=81 in logarithmic form.
Power and logarithm values
Power and logarithm valuesThe marked relationship is represented by the paired values $(3,8)$ and $(8,3)$. This graph shows only a window of the full domain.-202468-202468$y=2^x$$y=\log_2 x$xy

The marked relationship is represented by the paired values $(3,8)$ and $(8,3)$. This graph shows only a window of the full domain.

  1. Identify the parts
    The base is 3, the exponent is 4, and the result is 81. In logarithmic form, the result becomes the argument, while the exponent becomes the value of the logarithm.
    34=81  ⟹  log⁡381=43^4=81\;\Longrightarrow\;\log_3 81=4
  2. Check the conversion
    Read the logarithmic statement as a question about the exponent on 3. Since raising 3 to the fourth power gives 81, the statement is correct.
    34=813^4=81
Answer: log⁡381=4\log_3 81=4
Check: The base remains 3, the argument is 81, and the logarithm's value is the original exponent 4.

Worked example

Convert a logarithmic statement

Write log⁡5125=3\log_5 125=3 in exponential form.
Power and logarithm values
Power and logarithm valuesThe paired values illustrate that the exponent and result exchange roles. This graph shows only a window of the full domain.-202468-202468$y=2^x$$y=\log_2 x$xy

The paired values illustrate that the exponent and result exchange roles. This graph shows only a window of the full domain.

  1. Read the logarithm
    The base is 5. The argument is 125. The value 3 tells which exponent on 5 produces 125.
    53=1255^3=125
  2. Verify the power
    Multiplying 5 by itself three times gives 125, so the exponential statement correctly matches the original logarithm.
    5×5×5=1255\times5\times5=125
Answer: 53=1255^3=125
Check: The argument 125 is the result of the power, and 3 is its exponent.

Worked example

Find a variable argument

Solve log⁡2x=5\log_2 x=5 and check the restriction on xx.
Power and logarithm values
Power and logarithm valuesFor example, the value 32 corresponds to exponent 5 for base 2. This graph shows only a window of the full domain.048121620242832-202468$y=2^x$$y=\log_2 x$xy

For example, the value 32 corresponds to exponent 5 for base 2. This graph shows only a window of the full domain.

  1. Rewrite in exponential form
    The base is 2, the logarithm's value is 5, and the argument is xx. The equivalent power statement makes xx the result.
    25=x2^5=x
  2. Evaluate and check
    The fifth power of 2 is 32, so x=32x=32. A logarithm's argument must be positive. Since 32 is positive, the solution is allowed.
    x=32>0x=32>0
Answer: x=32x=32
Check: log⁡232=5\log_2 32=5 because 25=322^5=32.

Common mistakes and how to avoid them

Writing the exponent as the logarithm's argument.
Correction: The result of the power is the argument. The exponent is the value of the logarithm: bx=Nb^x=N matches log⁡bN=x\log_b N=x.
Changing the base during conversion.
Correction: Keep the base in the same role. In log⁡749=2\log_7 49=2, the matching power is 72=497^2=49.
Treating a negative argument as allowed.
Correction: A real logarithm requires a positive argument. For example, log⁡4(−16)\log_4(-16) is not defined as a real logarithm.
Assuming that log⁡bN\log_b N means bb multiplied by NN.
Correction: A logarithm gives an exponent. It asks which power of the base produces the argument.

Lesson summary

  • The statements bx=Nb^x=N and log⁡bN=x\log_b N=x express the same relationship.
  • The base stays the same; the exponent becomes the logarithm's value; the power's result becomes the argument.
  • For real logarithms, the base is positive and not 1, and the argument is positive.
  • Check a logarithmic statement by writing its equivalent power statement.

Check your understanding

Question 1

Which exponential statement is equivalent to log⁡464=3\log_4 64=3?
  1. 43=644^3=64
  2. 643=464^3=4
  3. 34=643^4=64
  4. 464=34^{64}=3
Show answer and explanation
43=644^3=64
The base is 4, the exponent is 3, and the result is 64.

Question 2

What is the logarithmic form of 72=497^2=49?
  1. log⁡749=2\log_7 49=2
  2. log⁡497=2\log_{49} 7=2
  3. log⁡27=49\log_2 7=49
  4. log⁡72=49\log_7 2=49
Show answer and explanation
log⁡749=2\log_7 49=2
The result 49 becomes the argument, and the exponent 2 becomes the value of the logarithm.

Question 3

Which statement is not defined as a real logarithm?
  1. log⁡39\log_3 9
  2. log⁡1/28\log_{1/2} 8
  3. log⁡15\log_1 5
  4. log⁡51\log_5 1
Show answer and explanation
log⁡15\log_1 5
A logarithm's base cannot equal 1. The other choices have positive bases other than 1 and positive arguments.

Key terms

Base
The number raised to a power. It is also the small number written with a logarithm.
Exponent
The number that tells which power of the base is used.
Argument
The number written after the base in a logarithm; it is the result produced by the matching power.
Equivalent statements
Statements written differently but expressing the same relationship.

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Published by DoAssignment. This reviewed lesson follows Alberta Grade 12 Mathematics (Math 30-1), expectation RF7. It is a study resource, not an official curriculum publication.

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