Yaadannoo boqonnaa fi gaaffii shaakalaa

Boqonnaa 3 Herrega kutaa 10: Hojiiwwan Exponential fi Logarithmic

Wanti fuula kana irra jiru hundi boqonnaa 3 kitaaba Herrega kutaa 10 tiif: wanta boqonnaan haguugu, yaadannoo gabaabaa, gaaffii shaakalaa deebii isaanii waliin, fi viidiyoo barnootaa bakka jirutti.

4
Kaardii: 10
Gaaffii shaakalaa boqonnaa 3: 116–185
Fuula kitaabaa boqonnaa 3

Boqonnaan 3 maal haguuga

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

Yaadannoo gabaabaa boqonnaa 3

Kaardiiwwan fuula Herrega kutaa 10 irratti boqonnaa kana jala galmaa'an. Tokko tokkoon bani.

Dogoggoroota Loogaariitimii BeekamooDogoggora baratame

Yoom akka itti fayyadamtu

Shallaggii loogaariitimii keessatti seera sirrii ta'uu mirkaneeffachuuf itti fayyadami.

Of eeggadhu

Ida'uu irratti loogaariitimii hin raabsinaa: $\log_a(M+N)$ fi $\log_a M + \log_a N$ tokko miti. Akkasumas $\log_a(MN)$ fi $(\log_a M)(\log_a N)$ adda adda.

Grade 10 Mathematics, fuula 117-186 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Giraafii Faankishinoota Eksipoonenshiyaalii fi LoogaariitimiiGabatee wabiiDomain · Range · Intercept · +3

Amala	Eksipoonenshiyaalii $f(x) = a^x$ ($a > 1$)	Loogaariitimii $f(x) = \log_a x$ ($a > 1$)
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 $y = x$ )

Yoom akka itti fayyadamtu

Doomeenii, reenzjii, qaxxaamura fi asimptootii faankishinoota kenne shallaguuf itti fayyadami.

Of eeggadhu

Faankishiniin eksipoonenshiyaalii sarara $y = 0$ hin tuqu, faankishiniin loogaariitimiis $x \le 0$ irratti hin hojjetu.

Grade 10 Mathematics, fuula 117-186 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Seerota EksipoonantiiGabatee wabiiProduct of powers · Quotient of powers · Power of a power · +5

Maqaa Seerichaa	Foorimulaa	Haala
Product of powers	$a^m \cdot a^n = a^{m+n}$	$a$ n real number dha
Quotient of powers	$\frac{a^m}{a^n} = a^{m-n}$	$a$ n 0 wajjin wal qixa miti
Power of a power	$(a^m)^n = a^{mn}$	$m, n$ n integers dha
Power of a product	$(ab)^n = a^n b^n$	$a, b$ n real numbers dha
Power of a quotient	(\frac{a}{b})^n = \frac{a^n}{b^n}	$b$ n 0 wajjin wal qixa miti
Zero exponent	$a^0 = 1$	$a$ n 0 wajjin wal qixa miti
Negative exponent	$a^{-n} = \frac{1}{a^n}$	$a$ n 0 wajjin wal qixa miti
Rational exponent	$a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$	$a > 0$ yookiin $n$ n odd dha

Yoom akka itti fayyadamtu

Himatamoota aljebiraa eksipoonantii qaban salphisuuf yookiin shallaguuf itti fayyadami.

Of eeggadhu

Mallattoo qoodaa keessaa eeggadhu: $(-3)^2 = 9$ garuu $-3^2 = -9$ dha. Akkasumas $0^0$ hiika hin qabu.

Grade 10 Mathematics, fuula 117-186 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Seerota LangaaritimiiGabatee wabiiSeera Product · Seera Quotient · Seera Power · +4

Maqaa Seerichaa	Foorimulaa	Haala
Seera Product	$\log_a(MN) = \log_a M + \log_a N$	$M, N > 0, a > 0, a$ 1 wajjin wal qixa kan hin taane
Seera Quotient	$\log_a(\frac{M}{N}) = \log_a M - \log_a N$	$M, N > 0, a > 0, a$ 1 wajjin wal qixa kan hin taane
Seera Power	$\log_a(M^r) = r \log_a M$	$M > 0, r$ real dha
Change of base	$\log_a M = \frac{\log_b M}{\log_b a}$	$a, b > 0$ , lachanuu 1 wajjin wal qixa kan hin taane
Log of base	$\log_a a = 1$	$a > 0, a$ 1 wajjin wal qixa kan hin taane
Log of one	$\log_a 1 = 0$	$a > 0, a$ 1 wajjin wal qixa kan hin taane
Inverse identity	$a^{\log_a M} = M$	$M > 0, a > 0, a$ 1 wajjin wal qixa kan hin taane

Yoom akka itti fayyadamtu

Himatamoota loogaariitimi bal'isuuf, walitti fiduuf, yookiin bu'aa isaanii shallaguuf itti fayyadami.

Of eeggadhu

Loogaariitimiin kan hojjetu lakkoofsota poozatiivii fi bu'uura 1 hin taane qofaaf.

Grade 10 Mathematics, fuula 117-186 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Gaaffii shaakalaa boqonnaa 3

Gaaffii filannoo 10 boqonnaa kana irratti barreeffaman, tokkoon tokkoon isaanii deebii fi sababa qabu. Deebii banuu dura yaali. Gaaffiin kun Ingiliffaa tiin.

Gaaffii 1

Evaluate -2³.

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

Deebii fi sababa agarsiisi

Deebii: A -8

Sababa

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.

Gaaffii 2

Find the value of log₂ 32.

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

Deebii fi sababa agarsiisi

Deebii: B 5

Sababa

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

Gaaffii 3

Simplify a³ × a².

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

Deebii fi sababa agarsiisi

Deebii: B a5

Sababa

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

Gaaffii 4

Evaluate 3^-2.

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

Deebii fi sababa agarsiisi

Deebii: A 1/9

Sababa

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

Gaaffii 5

Express log₃ 5 + log₃ 8 as a single logarithm.

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

Deebii fi sababa agarsiisi

Deebii: B log3 40

Sababa

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.

Gaaffii 6

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

Deebii fi sababa agarsiisi

Deebii: B 2

Sababa

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.

Gaaffii 7

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

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

Deebii fi sababa agarsiisi

Deebii: B 2

Sababa

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.

Gaaffii 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

Deebii fi sababa agarsiisi

Deebii: A 140.26 birr

Sababa

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.

Gaaffii 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

Deebii fi sababa agarsiisi

Deebii: A -2.45264

Sababa

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.

Gaaffii 10

Find the value of log_√2 4.