Logarithms answer the question “what power do I need?” Like a ruler that measures multiplicative steps instead of additive ones, logarithms turn products into sums and powers into products, and they are the key that unlocks every exponential equation.
Today's hook, If $2^x = 100$, you can't isolate $x$ by dividing or square-rooting. You need a new operation, one that undoes exponentiation. That operation is the logarithm, and by the end of this lesson you'll be able to evaluate it, find domains, and prove the inverse relationship.
0/5QUESTS
1
You’re here
Orient and recall
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
Worksheets
Practise this lesson
Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.
If $2^x = 100$, between which two whole numbers must $x$ lie? Lock in a guess before you read further, write the two numbers and your reasoning.
auto-saved
2
You’re here
The two moves
Work through the core explanation before applying it.
02
The two moves
+5 XP to read
There is really only one move in this entire lesson, converting between logarithmic and exponential form. Lock $\log_a x = y \iff a^y = x$ into muscle memory and every evaluation becomes routine.
The logarithm and the exponent are two sides of the same coin: $\log_a x = y$ says “the exponent is $y$”, while $a^y = x$ says “raising $a$ to that exponent gives $x$”. They are logically identical.
$\log_a x = y \iff a^y = x$
Special values
$\log_a 1 = 0$ and $\log_a a = 1$ for any valid base $a$. These are guaranteed by $a^0=1$ and $a^1=a$.
Domain constraint
The argument of a logarithm must be strictly positive: $x > 0$. You cannot take $\log$ of zero or a negative.
Inverse identity
$\log_a(a^x) = x$ and $a^{\log_a x} = x$. Logarithms and exponentials perfectly undo each other.
3
You’re here
What you'll master
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
03
What you'll master
Know
Key facts
The definition: $y = \log_a x$ means $a^y = x$
That $a > 0$, $a \neq 1$, and $x > 0$
Special cases: $\log_a 1 = 0$, $\log_a a = 1$
Understand
Concepts
Why logarithms and exponentials are inverse functions
Why the domain is restricted to $x > 0$
How common ($\log_{10}$) and natural ($\ln$) logs are special cases
Can do
Skills
Evaluate logarithmic expressions by converting to exponential form
Solve equations of the form $\log_a x = c$
Find the domain of $f(x) = \log_a(g(x))$
04
Key terms
LogarithmIf $a^y = x$ then $y = \log_a x$. The logarithm is the exponent needed to produce $x$ from base $a$.
Common logarithmLogarithm base 10: $\log_{10} x$, often written $\log x$.
Natural logarithmLogarithm base $e$: $\ln x = \log_e x$, where $e \approx 2.718$.
Inverse functionA function that reverses another; $f^{-1}(f(x)) = x$. Exponential and logarithmic functions are inverses of each other.
DomainFor $y = \log_a x$, the domain is $x > 0$; the argument must be positive.
RangeFor $y = \log_a x$, the range is all real numbers $(-\infty, \infty)$.
4
You’re here
What is a logarithm?
Work through the core explanation before applying it.
05
What is a logarithm?
core concept
A logarithm answers the question: "to what power must I raise the base to get this number?" The equation $\log_a x = y$ is completely equivalent to $a^y = x$. Switching between these two forms is the fundamental skill of this lesson.
$$\log_a x = y \iff a^y = x \qquad (a > 0,\ a \neq 1,\ x > 0)$$
Because $a^y > 0$ for all real $y$ when $a > 0$, the output of an exponential is always positive. This means when we invert to get $x = a^y$, the input $x$ must be positive, explaining why $\log_a 0$ and $\log_a(\text{negative})$ are undefined.
The inverse relationship in full. If $f(x) = a^x$ and $g(x) = \log_a x$, then $f(g(x)) = a^{\log_a x} = x$ and $g(f(x)) = \log_a(a^x) = x$. Both compositions give back $x$, confirming they are true inverses. Their graphs are reflections of each other across the line $y = x$.
Definition: $\log_a x = y \iff a^y = x$, where $a > 0$, $a \neq 1$, $x > 0$; Special values: $\log_a 1 = 0$ and $\log_a a = 1$ for any valid base
Pause, copy the logarithm definition ($\log_a x = y \iff a^y = x$) and the two special values ($\log_a 1 = 0$ and $\log_a a = 1$) into your book.
Did you get this? True or false: the expression $\log_3 9 = 2$ is equivalent to $3^2 = 9$.
Worked examples · 3 in a row, reveal as you go
5
You’re here
Work examples end to end
Follow the reasoning through complete worked solutions.
PROBLEM 1 · EVALUATE
Evaluate $\log_2 8$ and $\log_3 \dfrac{1}{9}$.
1
$\log_2 8$: find $y$ such that $2^y = 8$. Since $8 = 2^3$, we have $\log_2 8 = 3$.
$2^3 = 8$, so the exponent needed is 3.
2
$\log_3 \dfrac{1}{9}$: find $y$ such that $3^y = \dfrac{1}{9}$. Since $\dfrac{1}{9} = 3^{-2}$, we have $\log_3 \dfrac{1}{9} = -2$.
$3^{-2} = \frac{1}{9}$, so the exponent is $-2$. Negative logs are valid, they correspond to numbers between 0 and 1.
PROBLEM 2 · SOLVE
Solve $\log_5 x = 3$.
1
$\log_5 x = 3$ means $5^3 = x$
Convert from logarithmic form to exponential form.
For $\log_2(x - 3)$ to be defined, the argument must be positive: $x - 3 > 0$
The logarithm is only defined for positive inputs.
2
$x - 3 > 0 \implies x > 3$
Solve the inequality by adding 3 to both sides.
3
Domain: $(3, \infty)$
All real numbers strictly greater than 3. Note: $x = 3$ is excluded because $\log_2(0)$ is undefined.
Quick check: Which value equals $\log_4 64$?
Common errors · the 3 traps that cost marks
6
You’re here
Dodge the traps, then apply
Meet the mistakes that cost marks, then do it yourself.
Trap 01
Trying to evaluate $\log_a 0$ or $\log_a(\text{negative})$
$a^y$ is never zero or negative for real $y$ when $a > 0$. Therefore $\log_a x$ is only defined for $x > 0$. Writing $\log_2(-4)$ or $\log_3 0$ will always earn zero marks.
Trap 02
Confusing $\log_a(x + y)$ with $\log_a x + \log_a y$
$\log_a(x + y) \neq \log_a x + \log_a y$. The log of a sum is not the sum of logs. Only $\log_a(xy) = \log_a x + \log_a y$ is valid (log laws, next lesson).
Trap 03
Forgetting that $\log_a a^x = x$ for all real $x$
This identity is extremely useful for simplifying. $\log_2 2^5 = 5$ and $\ln e^3 = 3$. Students sometimes leave $\log_2 2^5$ unevaluated, always simplify using the inverse identity.
Fill in the blank: The domain of $f(x) = \ln(2x + 6)$ is $x >$ (enter as a number, e.g. -3).
Quick-fire practice · 5 problems
7
You’re here
Drill it, then lock it in
Run the quick drill and copy the summary into your book.
1
Evaluate $\log_2 16$ and $\log_{10} 1000$.
2
Evaluate $\log_3 \dfrac{1}{27}$.
3
Solve $\log_4 x = 2$.
4
Find the domain of $f(x) = \ln(2x + 1)$.
5
Simplify $\log_5 5^7$ and $e^{\ln 4}$.
Match each logarithm to its value. Drag or click to pair them.
$\log_2 32$
$\log_5 1$
$\log_3 \frac{1}{3}$
$\ln e^7$
$7$
$-1$
$0$
$5$
12
Revisit your thinking
Earlier you bracketed $x$ in $2^x = 100$. Since $2^6 = 64$ and $2^7 = 128$, the answer lives between 6 and 7. That value is exactly $\log_2 100 \approx 6.64$, the logarithm answers the question “what power of the base gives this number?” Now that you have the definition and the conversion technique, you can place any such value precisely.
auto-saved
1
You’re here
Multiple choice
Answer the drill bank and rate your confidence.
01
Multiple choice
+5 XP per correct · +25 XP all-correct
Pick your answer, then rate your confidence, that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
2
You’re here
Short answer
Write full responses, then check them against the model answers.
02
Short answer
ApplyBand 43 marks
Q1. Evaluate $\log_3 81$, $\log_2 \dfrac{1}{8}$, and $\ln e^4$. Show working for each. (3 marks)
auto-saved
ApplyBand 43 marks
Q2. Solve $\log_2(x + 1) = 4$ and state the domain of $f(x) = \log_2(x + 1)$. (3 marks)
auto-saved
AnalyseBand 53 marks
Q3. Show that $f(x) = 2^x$ and $g(x) = \log_2 x$ are inverse functions by verifying $f(g(x)) = x$ and $g(f(x)) = x$. (3 marks)
