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Connect exponential and logarithmic forms

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When converting the displayed equation to logarithmic form, what does the logarithm record?

\[a^{x}=n\]
Year 11, VCE Units 1 & 2, Chapter 11, 4 difficulty levels, No calculatorBuild a worksheet

Exponential and logarithmic forms

An exponential equation and a logarithmic equation can state the same relationship. The base stays the base, the exponent becomes the logarithm value, and the positive result becomes the argument

\[a^{x}=n\quad\Longleftrightarrow\quad \log_{a}\left(n\right)=x\]
  1. Keep the base valid A real logarithm has a positive base that is not 1 \(\displaystyle a>0,\quad a\ne 1\)
  2. Keep the argument positive The value inside a real logarithm must be positive \(\displaystyle n>0\)
  3. Read the logarithm as an exponent The logarithm gives the exponent that produces its positive argument from the base \(\displaystyle \log_{a}\left(n\right)=x\quad\Longleftrightarrow\quad a^{x}=n\)
  4. Use exact powers when possible If the argument is an exact power of the base, the logarithm value is the corresponding exponent \(\displaystyle \log_{2}\left(8\right)=3\quad\text{because}\quad 2^{3}=8\)

Worked example

Find the exact value of the logarithm below

\[\log_{4}\left(64\right)\]
  1. \(\displaystyle \log_{4}\left(64\right)=x\quad\Longleftrightarrow\quad 4^{x}=64\)Use the equivalence to turn the logarithm into an exponential equation
  2. \(\displaystyle 4^{3}=4\times 4\times 4=64\)The power of 4 that produces 64 has exponent 3
  3. \(\displaystyle x=3\)Therefore, the exact value is 3

Common mistakes

Swapping the logarithm roles
✕\(\displaystyle 2^{3}=8\quad\Longleftrightarrow\quad \log_{8}\left(2\right)=3\)✓\(\displaystyle 2^{3}=8\quad\Longleftrightarrow\quad \log_{2}\left(8\right)=3\)
The original base is the logarithm base and the positive result is the logarithm argument
Using a non-positive argument
✕\(\displaystyle \log_{2}\left(-8\right)=3\)✓\(\displaystyle \log_{2}\left(8\right)=3\)
A real logarithm is defined only when its argument is positive
Using 1 as a logarithm base
✕\(\displaystyle \log_{1}\left(8\right)=3\)✓\(\displaystyle \log_{2}\left(8\right)=3\)
A real logarithm base must be positive and different from 1

Difficulty levels

introA positive integer power is translated between exponential and logarithmic notation
standardEvaluate a negative or rational exponent while keeping the logarithm argument positive
challengeMixed conversions include a base between zero and one and non-integer exponents
examAn equivalent-statement choice is paired with a table of the base, exponent and positive argument

Questions about this topic

What does the logarithm of n to base a mean?+

It means the exponent x for which raising a to x gives n, provided a is positive and not 1 and n is positive

Why must a logarithm argument be positive?+

For a positive base, real powers always give positive values, so no real exponent produces a zero or negative logarithm argument

What happens when the base is between zero and one?+

The equivalence remains valid, but increasing the exponent makes the power smaller because the exponential function is decreasing

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 11: Exponentials and logarithms

In this chapter

Simplify and evaluate powersApply logarithm laws and inverse identitiesConnect exponential and logarithmic formsSolve exponential equations and inequalities exactlyConstruct exponential and logarithmic inverse pairsSolve exponential and logarithmic equationsAnalyse exponential and logarithmic graphsInterpret and use an exponential model