Exponents and logarithms AI HL

IB Mathematics AI HL · Unit 1

Number and Algebra · Lesson 1.5

Exponents and Logarithms

Use exponent laws to simplify expressions and logarithms to answer questions about powers.

Unit 1 theme · God’s love

✦ Wisdom

John 3:16 begins, “For God so loved the world, that he gave his only begotten Son.” God’s love is expressed through a gift, not through a number we could calculate. As we learn to represent quantities in different ways, we can also reflect on this clear invitation in Scripture: come to Jesus, receive his forgiveness, and trust him as your Savior and Lord.

🎯 Learning objectives

By the end of this lesson, you should be able to:

apply the laws of exponents with integer exponents;
simplify expressions containing negative and zero exponents;
recognise that a logarithm answers a question about an exponent;
evaluate base-10 and natural logarithms with technology;
solve introductory exponential equations; and
interpret logarithmic scales in context.

01 · What does an exponent tell us?

An exponent describes repeated multiplication. For example, three factors of 5 give:

$5^3=5\times5\times5=125.$

The base is 5 and the exponent is 3.

Worked example

Evaluate (−2)⁴ and −2⁴.

$(-2)^4=16$

$-2^4=-(2^4)=-16$

Parentheses matter. In the first expression, −2 is the base. In the second, the exponent applies only to 2.

02 · Multiplying and dividing powers

When multiplying powers with the same nonzero base, add their exponents. When dividing, subtract their exponents.

$\boxed{a^m a^n=a^{m+n}}$

$\boxed{\frac{a^m}{a^n}=a^{m-n}\qquad(a\ne0)}$

Worked example

Simplify each expression.

$5^3\times5^{-6}=5^{3-6}=\boxed{5^{-3}=\frac1{125}}$

$\frac{6^4}{6^3}=6^{4-3}=\boxed{6}$

Watch out: Exponents are added when multiplying powers with the same base. For example,

$2^3+2^4$

is an addition of numbers, not

$2^7.$

03 · Powers of powers and products

When raising a power to another power, multiply the exponents. An exponent outside parentheses applies to every factor inside the product.

$\boxed{(a^m)^n=a^{mn}}$

$\boxed{(ab)^n=a^n b^n}$

Worked example

Simplify the following expressions.

$(2^3)^{-4}=2^{-12}=\boxed{\frac1{4096}}$

$(2x)^4=2^4x^4=\boxed{16x^4}$

04 · Zero and negative exponents

For a nonzero base, an exponent of zero gives 1. A negative exponent gives a reciprocal.

$\boxed{a^0=1\qquad(a\ne0)}$

$\boxed{a^{-n}=\frac1{a^n}\qquad(a\ne0)}$

Worked example

Simplify the expression, assuming x is nonzero.

$2x^{-3}=\boxed{\frac{2}{x^3}}$

The negative exponent does not make the expression negative. It moves that factor into the denominator.

05 · What is a logarithm?

A logarithm asks: “To what power must the base be raised?” The following two statements mean the same thing:

$\boxed{a^x=b\quad\Longleftrightarrow\quad\log_a b=x}$

For a real logarithm, the base must be positive and cannot equal 1; the number inside the logarithm must be positive.

Worked example

Since

$10^3=1000,$

$\boxed{\log_{10}(1000)=3}.$

Also, since

$2^{-4}=\frac1{16},$

$\boxed{\log_2\left(\frac1{16}\right)=-4}.$

Read it aloud: “Log base 10 of 1000 is 3” means “10 to the power of 3 is 1000.”

06 · Base-10 and natural logarithms

The notation log commonly means logarithm to base 10. The notation ln means logarithm to base e, where e is approximately 2.71828.

$\log x=\log_{10}x$

$\ln x=\log_e x$

Worked example

Evaluate each logarithm to three decimal places.

$\log(0.0042)\approx\boxed{-2.377}$

$\ln(7.5)\approx\boxed{2.015}$

The first result is negative because 0.0042 lies between 0 and 1. The second is positive because 7.5 is greater than 1.

Calculator check: On a TI-84 Plus CE, use LOG for base 10 and LN for base e. Enter parentheses carefully.

07 · Solving an exponential equation

If a base raised to an unknown power equals a number, logarithms allow us to find that power. A calculator can use either log or ln; both give the same result when used consistently.

Worked example

Solve the equation below.

$3^{2t-1}=50$

$2t-1=\frac{\log 50}{\log 3}$

$t=\frac{1}{2}\left(1+\frac{\log 50}{\log 3}\right)$

$\boxed{t\approx2.280}$

Reasonableness check: When t is approximately 2.280, the exponent 2t − 1 is approximately 3.561. Since

$3^3=27$

and

$3^4=81,$

an answer of 50 is plausible.

08 · Logarithmic scales

A logarithmic scale compresses a very wide range of values. A fixed addition on a base-10 logarithmic scale corresponds to a fixed multiplication in the original quantity.

Worked example: acidity

The pH of a solution is modelled by the negative base-10 logarithm of its hydrogen ion concentration, measured in mol L⁻¹. Find the pH when the concentration is

$3.2\times10^{-5}\text{ mol L}^{-1}.$

$\mathrm{pH}=-\log(3.2\times10^{-5})$

$\boxed{\mathrm{pH}\approx4.49}$

A change of 1 pH unit corresponds to a factor of 10 in hydrogen ion concentration.

Other contexts: Logarithms also appear in sound intensity and earthquake measurements. Always read the exact scale formula supplied in a question.

✏️ IB-style Paper 1 practice

A graphic display calculator may be used. Show appropriate working. Give non-exact numerical answers to three significant figures unless stated otherwise.

Simplify

$7^4\times7^{-6}.$

Give your answer without a negative exponent.
[2 marks]
Simplify

$\frac{(3x^2)^3}{9x^{-1}},$

assuming x is nonzero.
[3 marks]
Write

$\log_4 64=3$

in exponential form. Then evaluate

$\log_5\left(\frac1{125}\right).$

[3 marks]
Evaluate

$\log(0.0042)$

and

$\ln(7.5)$

to three decimal places.
[2 marks]
Solve

$2^{3x}=100.$

Give your answer to three decimal places.
[3 marks]
Solve

$e^{2x}=12.$

Give your answer to three decimal places.
[3 marks]
A quantity is modelled by

$Q=750(1.08)^t,$

where t is
measured in years. Find the value of t when

$Q=1200.$

[4 marks]

Reveal Paper 1 solutions

$7^4\times7^{-6}=7^{-2} =\boxed{\frac1{49}}.$
$\frac{(3x^2)^3}{9x^{-1}} =\frac{27x^6}{9x^{-1}} =3x^{6-(-1)} =\boxed{3x^7}.$
The exponential statement is

$\boxed{4^3=64}.$

Since

$5^{-3}=\frac1{125},$

$\boxed{\log_5\left(\frac1{125}\right)=-3}.$
$\log(0.0042)\approx\boxed{-2.377},\qquad \ln(7.5)\approx\boxed{2.015}.$
$3x\log2=\log100$

$x=\frac{\log100}{3\log2} \approx\boxed{2.215}.$
$2x=\ln12$

$x=\frac{\ln12}{2} \approx\boxed{1.242}.$
$1.08^t=\frac{1200}{750}=1.6$

$t=\frac{\ln1.6}{\ln1.08} \approx\boxed{6.11\text{ years}}.$

🌍 Applications and advanced challenges

Investment target. An investment of €4800 earns
6% compound interest annually.

(a) Write a model for its value after t years.
[2 marks]

(b) Find when its modelled value reaches €7000.
[3 marks]

(c) If interest is credited only at the end of each full year,
find the first whole year by which the balance is at least €7000.
[2 marks]
Medicine model. A model assumes that 85% of a
medicine remains in the body after each hour. Initially, there are
120 mg.

(a) Write a model for the amount after t hours.
[2 marks]

(b) Find when the amount first falls below 12 mg according to
the continuous-time extension of this model.
[3 marks]

(c) If measurements are taken only after complete hours,
state the first whole hour at which less than 12 mg remains.
[2 marks]
Acidity. A water sample has hydrogen ion
concentration

$2.5\times10^{-5}\text{ mol L}^{-1}.$

(a) Calculate its pH using

$\mathrm{pH}=-\log[H^+].$

[2 marks]

(b) A second sample has pH 3.6. Find its hydrogen ion
concentration in mol L⁻¹.
[3 marks]

(c) Which sample has the greater hydrogen ion concentration?
[1 mark]
Sound intensity. Sound level is modelled by

$L=10\log\left(\frac{I}{I_0}\right),$

where L is in decibels and I₀ is
a reference intensity.

(a) Find

$I/I_0$

when

$L=65$

dB.
[3 marks]

(b) Explain why an increase of 10 dB corresponds to
multiplying the intensity by 10.
[3 marks]
Comparing two events. In a simplified earthquake
model, magnitude is

$M=\log\left(\frac{A}{A_0}\right),$

where A is measured amplitude and A₀
is a reference amplitude. Two events have magnitudes 6.2 and 4.7.
Find the ratio of their measured amplitudes.
[4 marks]

Reveal application solutions

(a)

$V=4800(1.06)^t.$

(b)

$7000=4800(1.06)^t,$

so

$t=\frac{\ln(7000/4800)}{\ln1.06} \approx\boxed{6.48\text{ years}}.$

(c) The first whole year is

$\boxed{7}.$
(a)

$A=120(0.85)^t.$

(b) Solve

$120(0.85)^t<12:$

$t>\frac{\ln0.1}{\ln0.85} \approx14.17\text{ hours}.$

(c) The first completed whole hour is

$\boxed{15\text{ hours}}.$

Model note: A real medicine’s concentration may not
follow the same percentage change at every moment.
(a)

$\mathrm{pH} =-\log(2.5\times10^{-5}) \approx\boxed{4.60}.$

(b)

$[H^+]=10^{-3.6} \approx\boxed{2.51\times10^{-4}\text{ mol L}^{-1}}.$

(c) The second sample has the greater
hydrogen ion concentration.
(a)

$65=10\log(I/I_0),$

so

$I/I_0=10^{6.5} \approx\boxed{3.16\times10^6}.$

(b) For two intensities, a sound-level increase of 10 dB
means the logarithms differ by 1. Since

$10^1=10,$

the intensity ratio is multiplied by

$\boxed{10}.$
The difference in magnitude is

$6.2-4.7=1.5.$

The corresponding amplitude ratio is

$10^{1.5}\approx\boxed{31.6}.$

The event of magnitude 6.2 has approximately 31.6 times
the measured amplitude in this simplified model.

✓ Lesson recap

Multiply powers of the same base by adding exponents; divide
by subtracting exponents.
A negative exponent produces a reciprocal:
$a^{-n}=\frac1{a^n}.$
A logarithm finds an exponent:
$a^x=b\quad\Longleftrightarrow\quad\log_a b=x.$
log uses base 10 and ln uses base e.
In an application, interpret the answer and check its units,
domain, and rounding.

Exponents and Logarithms

✦ Wisdom

🎯 Learning objectives

01 · What does an exponent tell us?

02 · Multiplying and dividing powers

03 · Powers of powers and products

04 · Zero and negative exponents

05 · What is a logarithm?

06 · Base-10 and natural logarithms

07 · Solving an exponential equation

08 · Logarithmic scales

✏️ IB-style Paper 1 practice

🌍 Applications and advanced challenges

✓ Lesson recap

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IB Mathematics: Applications and Interpretation HL 6 Sections80 LessonsLifetime

Curriculum

Exponents and logarithms AI HL

✦ Wisdom

🎯 Learning objectives

01 · What does an exponent tell us?

02 · Multiplying and dividing powers

03 · Powers of powers and products

04 · Zero and negative exponents

05 · What is a logarithm?

06 · Base-10 and natural logarithms

07 · Solving an exponential equation

08 · Logarithmic scales

✏️ IB-style Paper 1 practice

🌍 Applications and advanced challenges

✓ Lesson recap

Related

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