Unit notes and practice questions

Grade 10 Mathematics unit 3: Exponential and Logarithmic Functions

Everything here is for unit 3 of the Grade 10 Mathematics textbook: what the chapter covers, the short notes, practice questions with their answers, and the lesson video where one exists.

4
Cards: 10
Practice questions on unit 3: 116–185
The textbook pages for unit 3

What unit 3 covers

1.Exponents and Logarithms
2.The Exponential Functions and Their Graphs
3.The Logarithmic Functions and Their Graphs
4.Solving Exponential and Logarithmic Equations
5.Relation between Exponential and Logarithmic
6.Applications

Short notes for unit 3

The cards filed under this chapter on the Grade 10 Mathematics page. Open one at a time.

False Logarithmic IdentitiesCommon mistake

When you use it

Reach for this to verify log operations and prevent combining rules incorrectly.

Watch out

Do not distribute logarithms across addition: $\log_a(M+N)$ is not $\log_a M + \log_a N$ . Similarly, $\log_a(MN)$ is not $(\log_a M)(\log_a N)$ , and $\frac{\log_a M}{\log_a N}$ is not $\log_a(M-N)$ .

Drafted from Grade 10 Mathematics, pages 117-186, then checked twice before it went up

Graphs of Exponential and Logarithmic FunctionsReference tableDomain · Range · Intercept · +3

Feature	Exponential $f(x) = a^x$ ($a > 1$)	Logarithmic $f(x) = \log_a x$ ($a > 1$)
Domain	$(-\infty, \infty)$	$(0, \infty)$
Range	$(0, \infty)$	$(-\infty, \infty)$
Intercept	$y$ -intercept at $(0, 1)$	$x$ -intercept at $(1, 0)$
Asymptote	Horizontal asymptote $y = 0$	Vertical asymptote $x = 0$
Monotonicity	Increasing on $(-\infty, \infty)$	Increasing on $(0, \infty)$
Relationship	Inverse of $\log_a x$	Inverse of $a^x$ (reflection across $y = x$ )

When you use it

Use when finding the domain, range, intercepts, and asymptotes of exponential and logarithmic functions.

Watch out

The exponential function never touches $y = 0$ , and the logarithmic function is undefined for $x \le 0$ .

Drafted from Grade 10 Mathematics, pages 117-186, then checked twice before it went up

Laws of ExponentsReference tableProduct of powers · Quotient of powers · Power of a power · +5

Law Name	Formula	Condition
Product of powers	$a^m \cdot a^n = a^{m+n}$	$a$ is a real number
Quotient of powers	$\frac{a^m}{a^n} = a^{m-n}$	$a$ is not equal to 0
Power of a power	$(a^m)^n = a^{mn}$	$m, n$ are integers
Power of a product	$(ab)^n = a^n b^n$	$a, b$ are real numbers
Power of a quotient	(\frac{a}{b})^n = \frac{a^n}{b^n}	$b$ is not equal to 0
Zero exponent	$a^0 = 1$	$a$ is not equal to 0
Negative exponent	$a^{-n} = \frac{1}{a^n}$	$a$ is not equal to 0
Rational exponent	$a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$	$a > 0$ or $n$ is odd

When you use it

Use when simplifying algebraic expressions with integer or rational powers, or solving exponential expressions.

Watch out

The base in a negative expression matters: $(-3)^2 = 9$ but $-3^2 = -9$ . Also, $0^0$ is undefined.

Drafted from Grade 10 Mathematics, pages 117-186, then checked twice before it went up

Laws of LogarithmsReference tableProduct law · Quotient law · Power law · +4

Law Name	Formula	Condition
Product law	$\log_a(MN) = \log_a M + \log_a N$	$M, N > 0, a > 0, a$ not equal to 1
Quotient law	$\log_a(\frac{M}{N}) = \log_a M - \log_a N$	$M, N > 0, a > 0, a$ not equal to 1
Power law	$\log_a(M^r) = r \log_a M$	$M > 0, r$ is real
Change of base	$\log_a M = \frac{\log_b M}{\log_b a}$	$a, b > 0$ , both not equal to 1
Log of base	$\log_a a = 1$	$a > 0, a$ not equal to 1
Log of one	$\log_a 1 = 0$	$a > 0, a$ not equal to 1
Inverse identity	$a^{\log_a M} = M$	$M > 0, a > 0, a$ not equal to 1

When you use it

Use to expand, condense, or evaluate logarithmic expressions and to change the base of a logarithm.

Watch out

Logarithms are only defined for positive arguments and positive bases not equal to 1.

Drafted from Grade 10 Mathematics, pages 117-186, then checked twice before it went up

Practice questions on unit 3

10 multiple-choice questions written against this chapter, each with its answer and the reason. Try one before you open the answer.

Question 1

Evaluate -2³.

A.-8
B.8
C.-6
D.6

Show the answer and why

Answer: A -8

Why

In -2³ the base is only 2, because the minus sign is not inside the brackets. So 2³ = 2 × 2 × 2 = 8, and the answer is -8.

Question 2

Find the value of log₂ 32.

A.4
B.5
C.6
D.16

Show the answer and why

Answer: B 5

Why

A logarithm is an exponent. The question asks: 2 to what power gives 32? Since 2⁵ = 32, log₂ 32 = 5.

Question 3

Simplify a³ × a².

A.a6
B.a5
C.a9
D.a1

Show the answer and why

Answer: B a5

Why

When you multiply powers with the same base, you add the exponents (Law 1). So a³ × a² = a³⁺² = a⁵.

Question 4

Evaluate 3^-2.

A.1/9
B.-9
C.1/6
D.9

Show the answer and why

Answer: A 1/9

Why

For a negative exponent, a^-n = 1/aⁿ. So 3^-2 = 1/3² = 1/9.

Question 5

Express log₃ 5 + log₃ 8 as a single logarithm.

A.log3 13
B.log3 40
C.log3 3
D.log3 58

Show the answer and why

Answer: B log3 40

Why

The product law says that the sum of two logarithms with the same base is the logarithm of the product. So log₃ 5 + log₃ 8 = log₃ (5 × 8) = log₃ 40.

Question 6

Solve for x: 2^2x-1 = 8.

Show the answer and why

Answer: B 2

Why

Write 8 as a power of 2: 8 = 2³. So 2^2x-1 = 2³. If the bases are equal, the exponents are equal: 2x - 1 = 3, so 2x = 4 and x = 2.

Question 7

In the common logarithm of 345, what is the characteristic?

A.1
B.2
C.3
D.0.5378

Show the answer and why

Answer: B 2

Why

Write 345 in the form m × 10^c with 1 ≤ m < 10. Here 345 = 3.45 × 10². The characteristic is the power of 10, which is 2. The value 0.5378 is the mantissa, not the characteristic.

Question 8

A total of birr 100 is invested at an interest rate of 7% per year, compounded annually. Using A(t) = P(1 + r/n)^nt with n = 1 and t = 5, which amount is in the account after 5 years?

A.140.26 birr
B.135.00 birr
C.141.90 birr
D.107.00 birr

Show the answer and why

Answer: A 140.26 birr

Why

Substitute P = 100, r = 0.07, n = 1 and t = 5. Then A = 100(1.07)⁵ ≈ 140.26 birr. The value 141.90 birr is the answer for daily compounding, not annual.

Question 9

Solve for x using logarithms, given log 2 ≈ 0.3010 and log 3 ≈ 0.4771: 3^x+4 = 2^-x.

A.-2.45264
B.2.45264
C.-1.9084
D.-0.7781

Show the answer and why

Answer: A -2.45264

Why

Take the common logarithm of both sides and use the power law: (x + 4) log 3 = -x log 2. Expand to get x log 3 + x log 2 = -4 log 3, so x = -4 log 3 / (log 3 + log 2). Substituting, x ≈ -4(0.4771) / (0.4771 + 0.3010) = -1.9084 / 0.7781 ≈ -2.45264.

Question 10

Find the value of log_√2 4.