VertexMath · MHF4U · Free worksheet

Exponential and Logarithmic Functions Review

Try each problem first. You don’t need to finish everything.

Core path ~55 min Full sheet ~1 h 45 min

Recommended: K1–K5, then one A and one T problem

K = Knowledge & Understanding · A = Application · T = Thinking · Boss = Challenge

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Quick reference

Transformations of $y=a\cdot b^{k(x-d)}+c$ : factor $k$ out of the exponent before reading $d$ . The asymptote is $y=c$ .
Definition: $\log_b x=y \Leftrightarrow b^y=x$ . The argument must be positive, so $y=\log_b x$ has domain $(0,\infty)$ and asymptote $x=0$ .
Laws: $\log_b(mn)=\log_b m+\log_b n$ , $\log_b\dfrac{m}{n}=\log_b m-\log_b n$ , $\log_b m^n=n\log_b m$ , $\log_b m=\dfrac{\log m}{\log b}$ .
Models: growth of $r\%$ per period uses the base $1+\frac{r}{100}$ . A half-life $h$ gives $\left(\frac{1}{2}\right)^{t/h}$ , with $t$ and $h$ in the same units.
$\log(a+b)\ne\log a+\log b$ . After solving a log equation, check every root in the original equation: each argument must be positive.

Show your work on separate paper. Show your work in the dot-grid space.

Core Skills (K)

Exponential Graphs · Transformations · Equivalent Bases · Evaluating Logs · Log Laws

K1 From a Table

The table shows values of an exponential function $f(x)=a\cdot b^x$ .

$x$	$1$	$2$	$3$	$4$
$f(x)$	$20$	$5$	$1.25$	$0.3125$

a)
Determine $a$ and $b$ .
b)
State whether $f$ shows growth or decay. Then state its domain, its range and the equation of its asymptote.
c)
Evaluate $f(-2)$ and $f\left(\frac{1}{2}\right)$ .

Stuck? Peek at a hint

Divide each output by the one before it: that constant ratio is $b$ .

K2 Transformed Exponential

Let $g(x)=-3(2)^{2x-6}+4$ .

a)
Write the exponent in the form $k(x-d)$ . Then state $a$ , $k$ , $d$ and $c$ .
b)
State the equation of the asymptote, the domain and the range of $g$ .
c)
The point $(2,4)$ lies on $y=2^x$ . Determine its image on $g$ . Then determine the $y$ -intercept of $g$ .

Stuck? Peek at a hint

Factor the $2$ out of $2x-6$ first. A point $(x,y)$ maps to $\left(\frac{x}{k}+d,\ ay+c\right)$ .

K3 Same Base

Solve each equation by writing both sides as powers of the same base.

a)
$9^{x+1}=27^{x-1}$
b)
$4^{x}=\dfrac{1}{8\sqrt{2}}$
c)
$2^{x+3}-2^{x}=56$

Stuck? Peek at a hint

Write $9$ , $27$ , $4$ and $8\sqrt{2}$ as powers of $3$ or $2$ . In (c), $2^{x+3}=8\cdot2^x$ .

K4 Evaluating Logarithms

a)
Evaluate $\log_2\frac{1}{32}$ , $\log_8 4$ and $\ln\sqrt{e}$ .
b)
Solve $\log_x 125=3$ and $\log_6 x=-2$ .
c)
The graph of $y=\log_b x$ passes through the marked point. Determine $b$ , then state the domain and range of $y=b^x$ . Each grid square is 1 unit.

Stuck? Peek at a hint

Rewrite each logarithm as a question: "the base to what power gives the argument?"

K5 Laws of Logarithms

a)
Evaluate $\log_6 4+\log_6 9$ and $\log_3 54-\log_3 2$ .
b)
Write as a single logarithm: $2\log x+\frac{1}{2}\log y-3\log z$ .
c)
Evaluate $\dfrac{\log_2 64}{\log_2 8}$ and $\log_2 64-\log_2 8$ .

Stuck? Peek at a hint

Combine first with the product or quotient law, then evaluate. Move each coefficient into the argument as a power.

Real World (A)

Growth Models · Log Scales · Solving in Context

A1 Petri Dish

A lab culture starts with 500 bacteria and doubles every 20 minutes.

a)
Write $N(t)$ , the number of bacteria after $t$ hours.
b)
Find the number of bacteria after 4 hours.
c)
The dish can hold about $10^9$ bacteria. When does the model reach this? Then evaluate $N(24)$ and say what it tells you about the model.

Stuck? Peek at a hint

Count how many 20-minute periods fit in one hour.

A2 Leaf Blowers

The sound level, in decibels, of a sound with intensity $I$ (in W/m²) is $L=10\log\left(\dfrac{I}{I_0}\right)$ , where $I_0=10^{-12}$ W/m².

a)
A conversation has intensity $10^{-6}$ W/m². Find its sound level.
b)
A leaf blower measures 96 dB. How many times as intense as the conversation is it?
c)
Two identical 96 dB leaf blowers run side by side. Their intensities add. Find the combined sound level.

Stuck? Peek at a hint

Subtract the two levels, divide by 10, and use the result as a power of 10.

A3 Cooling Tea

A cup of tea is left on a counter in a 20 °C room. Its temperature, in °C, after $t$ minutes is $T(t)=20+70(0.5)^{t/12}$ .

a)
Find the temperature of the tea after 30 minutes.
b)
When is the tea at 50 °C?
c)
When does the tea reach 18 °C? Interpret your result.

Stuck? Peek at a hint

Isolate $(0.5)^{t/12}$ on one side before taking logarithms.

Think It Through (T)

Building Log Functions · Error Analysis

T1 Build the Log

Each part asks for the equation of a transformed logarithmic function.

a)
The graph is a transformation of $y=\log_2 x$ . Determine its equation in the form $y=a\log_2(x-d)+c$ . Each grid square is 1 unit.
b)
Write the equation of a logarithmic function with vertical asymptote $x=5$ , domain $(-\infty,5)$ and $x$ -intercept $4$ .
c)
A function $y=a\log_3(x-d)+c$ has vertical asymptote $x=-2$ and passes through $(1,5)$ and $(7,7)$ . Determine $a$ , $d$ and $c$ .

Stuck? Peek at a hint

Start from the asymptote: it gives $d$ . Then pick points where the argument is $1$ , $2$ or $4$ (or a power of $3$ in (c)).

T2 Spot the Flaw

A student solves $\log_2(x+2)+\log_2(x-4)=4$ :
"Product law: $\log_2\big((x+2)(x-4)\big)=4$ , so $(x+2)(x-4)=16$ . Then $x^2-2x-24=0$ and $(x-6)(x+4)=0$ . Both $x=6$ and $x=-4$ are solutions."

a)
Check each claimed solution in the original equation. Which one fails, and why?
b)
State the domain of the original equation and the domain of $\log_2\big((x+2)(x-4)\big)=4$ . Explain how this lets the wrong root in.
c)
Another student writes $\log_2(x+2)+\log_2(x-4)=\log_2(2x-2)$ . Use $x=6$ to show this is false in general. Then find the only $x$ for which it is true.

Stuck? Peek at a hint

Substitute each value into every logarithm of the original equation, one at a time.

Boss Level

Optional challenge

A Chain of Logs

Use the change of base formula throughout.

a)
Evaluate $\log_2 3\cdot\log_3 4$ .
b)
Evaluate $\log_2 3\cdot\log_3 4\cdot\log_4 5\cdots\log_{63}64$ .
c)
Solve $\log_2 x+\log_4 x+\log_{16}x=7$ .

Stuck? Peek at a hint

Write each factor as $\dfrac{\log m}{\log b}$ and look for cancellations. In (c), express every term with $\log_2 x$ .

Answer key

Final answers only. Try first, then check.

Core Skills

a) $a=80$ , $b=\frac{1}{4}$
b) Decay; $D=(-\infty,\infty)$ , $R=(0,\infty)$ ; asymptote $y=0$
c) $f(-2)=1280$ ; $f\left(\frac{1}{2}\right)=40$

a) $g(x)=-3(2)^{2(x-3)}+4$ ; $a=-3$ , $k=2$ , $d=3$ , $c=4$
b) $y=4$ ; $D=(-\infty,\infty)$ , $R=(-\infty,4)$
c) $(4,-8)$ ; $y$ -intercept $\frac{253}{64}\approx3.95$

a) $x=5$
b) $x=-\frac{7}{4}$
c) $x=3$

a) $-5$ ; $\frac{2}{3}$ ; $\frac{1}{2}$
b) $x=5$ ; $x=\frac{1}{36}$
c) $b=3$ ; $y=3^x$ : $D=(-\infty,\infty)$ , $R=(0,\infty)$

a) $2$ ; $3$
b) $\log\dfrac{x^2\sqrt{y}}{z^3}$
c) $2$ ; $3$

Real World

a) $N(t)=500(2)^{3t}$
b) $2\,048\,000$ bacteria
c) After about $7.0$ h; $N(24)\approx2.4\times10^{24}$ , far more than the dish can hold, so the model is valid only for about the first 7 hours: growth must level off.

a) $60$ dB
b) $10^{3.6}\approx3981$ times (about 4000 times)
c) $96+10\log2\approx99.0$ dB, not 192 dB

a) $\approx32.4$ °C
b) $t=\dfrac{12\log(3/7)}{\log0.5}\approx14.7$ minutes
c) Never: $(0.5)^{t/12}=-\frac{2}{70}$ has no solution. The tea approaches room temperature, 20 °C (the asymptote), and cannot cool below it.

Think It Through

a) $y=-\log_2(x+3)+1$
b) e.g. $y=\log(5-x)$ , which is $y=\log(-(x-5))$
c) $a=2$ , $d=-2$ , $c=3$ : $y=2\log_3(x+2)+3$

a) $x=6$ works: $\log_2 8+\log_2 2=3+1=4$ . $x=-4$ fails: $\log_2(-2)$ is undefined, so it is extraneous.
b) Original: $(4,\infty)$ . Combined: $(-\infty,-2)\cup(4,\infty)$ . The product law widened the domain, so the combined equation accepts $x=-4$ .
c) At $x=6$ : left side $4$ , right side $\log_2 10\approx3.32$ . True only at $x=2+\sqrt{10}$ .

Boss Level

Boss

a) $2$
b) $6$
c) $x=16$

Free cheat sheet

Exponential and Logarithmic Functions Cheat Sheet

The whole unit on one page: every rule, a worked example for each idea, and the traps that cost marks.

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Most testedSolve $\log_2x+\log_2(x-2)=3$ . (card 7)

1Exponential functions

Use $y=b^x$ , $b>0$ , $b\ne1$ : domain $\mathbb{R}$ , range $y>0$ , asymptote $y=0$ , $y$ -intercept $1$ . $b>1$ grows, $0<b<1$ decays. A table has constant ratios.

Trap $2^{-3}=1/8$ , still positive: the range is $y>0$ and the curve never reaches $y=0$ .

2Exponential transformations

$y=a\,b^{k(x-d)}+c$ : asymptote $y=c$

UseFactor $k$ out of the exponent first. Map $(x,y)\to(x/k+d,\ ay+c)$ .

Trap $2^{2x-6}=2^{2(x-3)}$ : the shift is $3$ right, not $6$ .

3Equal bases

UseWrite both sides as powers of the same base, then set the exponents equal. A difference like $b^{x+3}-b^x$ : factor out $b^x$ .

Example $4^{x+1}=8^x$ : $2^{2x+2}=2^{3x}$ , so $x=2$ .

Trap $2^{x+3}-2^x=2^x(2^3-1)=7\cdot2^x$ , not $2^3$ .

4Logarithms

$\log_bx=y\iff b^y=x$

Use $y=\log_bx$ : domain $x>0$ , asymptote $x=0$ , $x$ -intercept $1$ . It is the inverse of $b^x$ . $\log$ means base $10$ ; $\ln$ means base $e$ .

Trap $\log_x125=3$ means $x^3=125$ , so $x=5$ , not $125^3$ .

5Log transformations

$y=a\log_b[k(x-d)]+c$ : asymptote $x=d$

UseFactor $k$ out first. Domain: solve $k(x-d)>0$ . The $x$ -intercept comes from $a\log_b[k(x-d)]+c=0$ .

Trap $y=\log(5-x)$ needs $5-x>0$ : domain $x<5$ , not $x>5$ .

6Laws of logarithms

$\log_b(mn)=\log_bm+\log_bn$

$\log_b\dfrac mn=\log_bm-\log_bn$

$\log_bm^n=n\log_bm\qquad\log_bm=\dfrac{\log m}{\log b}$

Trap $\log u+\log v=\log(uv)$ , never $\log(u+v)$ .

Trap $\log_264\div\log_28=6\div3=2$ : a quotient of logs is not the log of a quotient.

7Solving with logarithms

UseExponential: isolate the power, then take $\log$ of both sides. Log equation: condense, convert to exponential form, then check every root; each argument must be positive.

Example $\log_2x+\log_2(x-2)=3$ : $x(x-2)=8$ , so $x=4$ ( $x=-2$ is rejected).

TrapIsolate first: $20(1.05)^t=50$ becomes $1.05^t=2.5$ , so $t=\log2.5/\log1.05\approx18.8$ .

8Models and log scales

$A=A_0(1+r)^t\qquad A=A_0(2)^{t/d}$

$A=A_0\left(\dfrac12\right)^{t/h}$

Use $t$ , $d$ and $h$ in the same units. Log scales: $+1$ in pH or magnitude, or $+10$ dB, means $\times10$ . Instantaneous rate: shrink the interval.

Trap $96$ dB is $10^{3.6}\approx3981$ times as intense as $60$ dB, not $1.6$ times.

Which strategy?

Both sides share a base Equate exponents

Bases don't match Take logs

Logs on one side Condense, convert

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