Secondary 4 Logarithmic Functions Worksheet

All 6 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Finding the Rule of a Logarithmic Function

   The function $f(x)=a{\log }_2(x-1)$ passes through the point $(9,6)$.

   1. Find the value of $a$.
   2. Using this rule, compute $f(5)$ and state the equation of the vertical asymptote.
2. Q2Graphing the Logarithmic Function

   Sketch $f(x)={\log }_2(x-1)$ on the grid, using at least three exact points. Give the equation of the vertical asymptote, the domain, the range and the zero.

   A blank Cartesian grid for this question is on the printable PDF.

3. Q3Properties of Logarithmic Functions

   Evaluate each expression without a calculator, naming the logarithm property used.

   1. ${\log }_{3}81$
   2. ${\log }_{5}100-{\log }_{5}4$
   3. $2{\log }_{2}6-{\log }_{2}9$
   4. ${\log }_{7}1+{\log }_{4}4$
4. Q4The Inverse of the Logarithmic Function

   Let $f(x)={\log }_3(x-2)+1$. Find the rule of $f^{-1}$, its domain and its range.

5. Q5The Logarithmic Function

   The logarithm is defined by ${\log }_c(x)=y⟺c^y=x$.

   1. State the conditions the base $c$ must satisfy for $f(x)={\log }_c(x)$ to be a logarithmic function, and say why each excluded value fails.
   2. Use the definition to explain why ${\log }_c(-9)$ does not exist for any admissible base, and deduce the domain and the asymptote of $f$.
6. Q6The Role of the Parameters in a Logarithmic Function

   Consider $f(x)=2{\log }_3(x-4)-4$.

   1. State the domain and the equation of the vertical asymptote.
   2. Evaluate $f(85)$.
   3. Find the zero of $f$.

The 5 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

One definition does most of the work

Everything on this sheet is this single line, applied in one direction or the other:

Read it in words: a logarithm is an exponent. "Log base c of x" asks the question "what power do I put on c to get x?" Q5 asks for the definition itself and for its consequences, so this line is not background — it is the answer.

Three facts drop straight out of it and are worth knowing by reflex, because they turn up in every evaluation question: log_c 1 = 0 (any base to the power 0 is 1), log_c c = 1, and log_c(cⁿ) = n.

Why the base must be positive and not 1

Q5(a) asks you to state the conditions and say why each excluded value fails. Each exclusion has its own reason and a generic "it doesn't work" earns nothing.

• A negative base makes c^y undefined for many exponents — a power of one half is a square root, and there is no real square root of a negative number. The rule would have gaps rather than a domain.
• Base 1 gives 1^y = 1 for every y. So the equation 1^y = x has no solution at all when x ≠ 1, and infinitely many when x = 1. Either way there is no function to define.
• Base 0 gives 0 for every positive exponent and is undefined for the rest, so it can never produce a strictly positive x.

The domain and the asymptote live in the bracket

Q1, Q2 and Q6 all ask for a domain or an asymptote, and all three are answered the same way: set the argument greater than zero and solve.

The asymptote sits where the bracket is zero — not at the number that appears inside it. For an argument written as (x + 3) the bracket vanishes at x = −3, so h = −3 and the asymptote is x = −3. Rewrite the argument as a subtraction before reading anything out of it.

What does not restrict anything: a and k sit outside the logarithm. They stretch, reflect and shift the curve vertically, but they cannot change which inputs are allowed. And the range of a logarithmic function is always ℝ — the curve keeps climbing (or falling) without bound, just very slowly. Domain restricted, range unrestricted: the exact mirror image of an exponential function.

Finding a rule from a point

Q1 gives one point and asks for the coefficient. The technique is to exploit the point whose argument is an exact power of the base, because then the logarithm evaluates to a whole number and the unknown falls out in one line.

A worked example of the same shape: for f(x) = a·log₅(x + 1) through (24, −6), the argument is 25, which is 5², so the logarithm is 2 and the equation is 2a = −6, giving a = −3.

When more parameters are unknown, take them in this order: the asymptote gives h immediately; a point where the argument equals 1 gives k immediately, because log_c 1 = 0 wipes out the a term; and any remaining point then gives a. Choosing that order turns a three-unknown problem into three one-line problems.

The laws — and the law that does not exist

Q3 asks you to evaluate several expressions exactly and to name the property used. The name is part of the mark, so learn them as named rules:

The working method is always the same: use the laws to collapse the expression into a single logarithm whose argument is an exact power of the base, then read the exponent. For instance 3·log₂2 + log₂4 becomes log₂8 + log₂4 by the power law, then log₂32 by the product law, and 32 is 2⁵, so the value is 5. Note the two named steps — that is the shape of a full answer.

Solving a logarithmic equation

Q6 finishes with a zero to find, and the order of operations is where these go wrong.

Step 4 is not a formality. Combining two logarithms into one enlarges the set of values the equation makes sense for, so the combined equation can have genuine solutions that the original equation cannot accept. They are not arithmetic mistakes — they are correct solutions to a different equation, and the only way to catch them is to test them against the original arguments.

Graphing: choose x so the argument is a power of the base

Q2 asks for exact points, and the way to get them is to work backwards from the values you want. If the argument equals 1 the logarithm is 0 — that is the zero of the function. If it equals the base, the logarithm is 1; the base squared gives 2; the reciprocal of the base gives −1. Solve each of those for x and you have four exact points with no decimals anywhere.

Draw the vertical asymptote as a dashed line before the curve. The curve hugs it, plunging without bound on that side, and never crosses it. On the other side it keeps rising forever — slowly, but with no ceiling, which is what "range ℝ" looks like. The function increases when a > 0 and the base is greater than 1; a negative a reflects it and makes it decreasing.

The inverse: back to an exponential

Q4 asks for the inverse of a logarithmic rule. Swap x and y, then undo in reverse order: the constant first, then the coefficient, and only then the logarithm itself — which is undone by raising the base to both sides.

Then read the properties off the swap instead of recomputing them. The domain of the inverse is the range of f, which is all of ℝ; the range of the inverse is the domain of f, which is the half-line the logarithm allowed; and the vertical asymptote x = h becomes the horizontal asymptote y = h of the exponential inverse.

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Getting the most out of it

Read every logarithm out loud as a question

"Log base 3 of 81" is not a symbol to manipulate, it is the question "3 to what power gives 81?". Saying it that way converts most evaluation questions into mental arithmetic and makes the definition impossible to forget under exam pressure.

Learn the powers of 2, 3 and 5

Up to about a thousand, plus their reciprocals. Every exact-evaluation question on this sheet is built so the argument is a power of the base, so recognising it instantly is the difference between a one-line answer and a stuck question.

Write the domain before the first line of algebra

On any equation containing a logarithm, note "argument > 0" at the top of your work and solve it. Doing it first turns the rejection at the end from an afterthought into a confirmation, and it is where the marks on these questions actually sit.

Common questions