What a Logarithm Is
A logarithm is not a new operation. It is an index you have been asked for instead of given, and once you can see that, every rule that follows is one you already know.
You already know that $2^5 = 32$. Now answer the reverse question: $2$ raised to what power gives $32$? You can do that one in your head. Try a harder one: $2$ raised to what power gives $100$? You cannot say exactly, but you can say it is between two whole numbers. Which two, and how do you know?
A logarithm is an index. When you write $\log_a b$ you are asking one question and one question only: what power do I put on $a$ to get $b$? The answer to that question is the logarithm. Nothing else is going on.
$$\log_a b = c \quad\Longleftrightarrow\quad a^c = b$$
Read the left-hand side as "the logarithm of $b$ to the base $a$". Read the right-hand side as "$a$ to the power $c$ is $b$". They say the same thing about the same three numbers, and the arrow points both ways. Every logarithm question you will ever meet is one of these two forms in disguise.
Know
- That $\log_a b$ is the index to which the base $a$ must be raised to give $b$
- The equivalence $a^c = b \iff \log_a b = c$
- That the base must satisfy $a > 0$ and $a \ne 1$, and that $b > 0$
Understand
- Why a logarithm is a question about an index rather than a new operation
- Why a logarithm of zero or of a negative number does not exist
Can Do
- Translate a statement in index form into logarithmic form, and back
- Evaluate simple logarithms by inspection, such as $\log_2 32$
- Estimate a logarithm that is not a whole number by trapping it between two that are
Say $\log_2 32$ as "log, base two, of thirty-two". Saying it properly matters, because the words tell you what to do: log means "the index", base two tells you what carries the index, and of thirty-two tells you what the index has to produce.
So $\log_2 32$ is shorthand for the sentence "the index that turns 2 into 32". That index is $5$, because $2^5 = 32$. Therefore $\log_2 32 = 5$.
When no base is written, as in $\log 1000$, the base is taken to be $10$. This is the common logarithm, and it is the one your calculator's log button uses.
Translation is mechanical once you know where each number goes. Start from index form:
$$a^c = b \quad\Longrightarrow\quad \log_a b = c$$
The base stays the base. The index becomes the answer. The result becomes the argument.
| Index form | Logarithmic form |
|---|---|
| $3^4 = 81$ | $\log_3 81 = 4$ |
| $10^{-2} = 0.01$ | $\log_{10} 0.01 = -2$ |
| $5^0 = 1$ | $\log_5 1 = 0$ |
| $16^{1/2} = 4$ | $\log_{16} 4 = \dfrac{1}{2}$ |
Notice the third and fourth rows. A logarithm can be zero, negative or fractional. It is an index, and indices can be any of those.
For a logarithm with friendly numbers, do not reach for a calculator. Ask the question the notation is asking.
To find $\log_3 81$, ask: 3 to what power is 81? Count up: $3, 9, 27, 81$. That is four steps, so the answer is $4$.
To find $\log_4 \dfrac{1}{16}$, ask: 4 to what power is $\dfrac{1}{16}$? Since $4^2 = 16$, you need $4^{-2}$, so the answer is $-2$.
Two restrictions come straight from the definition, and neither is arbitrary.
The argument must be positive. Ask for $\log_2 (-8)$ and you are asking what power of $2$ gives $-8$. Every power of a positive number is positive, so there is no answer. The same applies to $\log_2 0$: no power of $2$ ever reaches zero, it only gets closer.
The base must be positive and not $1$. If $a = 1$ then $1^c = 1$ for every $c$, so $\log_1 5$ has no answer and $\log_1 1$ has infinitely many. Neither is useful, so the base $1$ is excluded.
Written formally: $\log_a b$ is defined when $a > 0$, $a \ne 1$ and $b > 0$.
Watch Me Solve It · 3 examples
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1Identify the three parts$\text{base} = 7, \quad \text{index} = 3, \quad \text{result} = 343$The base carries the power; the result is what the power produces.
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2Place them in log form$\log_{\text{base}} (\text{result}) = \text{index}$The base stays low, the result becomes the argument, the index becomes the answer.
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3Write it out$\log_7 343 = 3$
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4Check by translating back$7^3 = 343 \;\checkmark$Every translation should survive a round trip.
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1Name the parts$\text{base} = 5, \quad \text{argument} = 0.04, \quad \text{logarithm} = -2$The logarithm is the index you are being handed.
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2Apply the equivalence$\log_a b = c \iff a^c = b$
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3Substitute$5^{-2} = 0.04$
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4Check the arithmetic$5^{-2} = \dfrac{1}{5^2} = \dfrac{1}{25} = 0.04 \;\checkmark$A negative index gives a reciprocal, which is why the argument is less than 1.
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1Look for a shared base$9 = 3^2, \qquad 27 = 3^3$Neither number is a whole-number power of the other, but both are powers of 3.
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2Set the logarithm equal to the unknown index$\log_9 27 = x \iff 9^x = 27$Turn the question back into index form, where you have rules.
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3Rewrite both sides to base 3$(3^2)^x = 3^3$$3^{2x} = 3^3$
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4Equate the indices and solve$2x = 3$$x = \dfrac{3}{2}$With equal bases, equal values force equal indices.
Brain Trainer · 4 problems
Four quick problems. Work each one, then reveal the answer.
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1 Write $2^6 = 64$ in logarithmic form.
The base is $2$, the index is $6$, the result is $64$.$\log_2 64 = 6$ -
2 Write $\log_4 64 = 3$ in index form.
The base is $4$ and the logarithm $3$ is the index.$4^3 = 64$ -
3 Evaluate $\log_5 125$.
$5^3 = 125$, so the index is $3$.$3$ -
4 Between which two consecutive whole numbers does $\log_3 50$ lie?
$3^3 = 27$ and $3^4 = 81$, and $27 < 50 < 81$.Between $3$ and $4$
Multiple Choice · 5 questions
The statement $\log_a b = c$ is equivalent to:
The value of $\log_2 128$ is:
Which statement is written correctly in logarithmic form?
Which of these is undefined?
Without a calculator, $\log_4 60$ lies between:
Short Answer · 3 questions
(b) Write $\log_{12} 144 = 2$ in index form.
(c) Explain, in one sentence, why the two forms carry the same information.
(a) $\log_3 243$
(b) $\log_8 \dfrac{1}{64}$
(c) $\log_{25} 125$
(a) Explain why this is wrong.
(b) Evaluate $\log_2 \dfrac{1}{16}$ and explain what the negative answer actually means.
(a) Write an expression for the thickness, in millimetres, after $n$ folds.
(b) Write, in logarithmic form, the statement that gives the number of folds needed to reach $1$ metre.
(c) Trap that number of folds between two consecutive whole numbers, without a calculator.
Definition
$\log_a b = c \iff a^c = b$
In words
The index that turns the base into the argument
Restrictions
$a > 0$, $a \ne 1$, $b > 0$
Check
Translate back to index form and test it
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