We read your photo as:

Given that log10 2 = 0.3010 and log10 3 = 0.4771, evaluate the following without using mathematic tables: a) log10 18 b) log10 √0.6

Mathematics

ScanSolve AI Answer

Answers

2 parts
a
1.2552
b
-0.1110

Step-by-step solution

11 steps

Hapa kuna jinsi ya kutatua swali hili kwa kutumia sheria za logariti na thamani ulizopewa:

Tunapewa:
log⁡102=0.3010\log_{10} 2 = 0.3010
log⁡103=0.4771\log_{10} 3 = 0.4771

Part a· 4 steps

Tathmini log⁡1018\log_{10} 18

  1. Andika 18 kama bidhaa ya nambari kuu (prime factors).

    18=2×9=2×3218 = 2 \times 9 = 2 \times 3^2

  2. Tumia sheria ya logariti ya kuzidisha (log⁡b(MN)=log⁡bM+log⁡bN\log_b (MN) = \log_b M + \log_b N) na sheria ya nguvu (log⁡b(Mp)=plog⁡bM\log_b (M^p) = p \log_b M).

    log⁡1018=log⁡10(2×32)=log⁡102+log⁡1032=log⁡102+2log⁡103\log_{10} 18 = \log_{10} (2 \times 3^2) = \log_{10} 2 + \log_{10} 3^2 = \log_{10} 2 + 2 \log_{10} 3

  3. Badilisha thamani ulizopewa.

    log⁡1018=0.3010+2×0.4771\log_{10} 18 = 0.3010 + 2 \times 0.4771

  4. Fanya hesabu.

    log⁡1018=0.3010+0.9542=1.2552\log_{10} 18 = 0.3010 + 0.9542 = 1.2552

    Jibu la mwisho kwa a) ni:

    1.2552\boxed{1.2552}

Part b-0.1110

Tathmini log⁡100.6\log_{10} \sqrt{0.6}

  1. Andika upya mzizi wa mraba kama nguvu na 0.6 kama sehemu.

    log⁡100.6=log⁡10(0.6)1/2=log⁡10(610)1/2=log⁡10(35)1/2\log_{10} \sqrt{0.6} = \log_{10} (0.6)^{1/2} = \log_{10} \left(\frac{6}{10}\right)^{1/2} = \log_{10} \left(\frac{3}{5}\right)^{1/2}

  2. Tumia sheria ya nguvu ya logariti (log⁡b(Mp)=plog⁡bM\log_b (M^p) = p \log_b M).

    log⁡10(35)1/2=12log⁡10(35)\log_{10} \left(\frac{3}{5}\right)^{1/2} = \frac{1}{2} \log_{10} \left(\frac{3}{5}\right)

  3. Tumia sheria ya logariti ya kugawanya (log⁡b(MN)=log⁡bM−log⁡bN\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N).

    12log⁡10(35)=12(log⁡103−log⁡105)\frac{1}{2} \log_{10} \left(\frac{3}{5}\right) = \frac{1}{2} (\log_{10} 3 - \log_{10} 5)

  4. Eleza log⁡105\log_{10} 5 kwa kutumia log⁡1010\log_{10} 10 na log⁡102\log_{10} 2.

    Kumbuka log⁡1010=1\log_{10} 10 = 1.

    log⁡105=log⁡10(102)=log⁡1010−log⁡102=1−log⁡102\log_{10} 5 = \log_{10} \left(\frac{10}{2}\right) = \log_{10} 10 - \log_{10} 2 = 1 - \log_{10} 2

  5. Badilisha usemi wa log⁡105\log_{10} 5 kwenye mlinganyo wa awali.

    12(log⁡103−(1−log⁡102))=12(log⁡103−1+log⁡102)\frac{1}{2} (\log_{10} 3 - (1 - \log_{10} 2)) = \frac{1}{2} (\log_{10} 3 - 1 + \log_{10} 2)

  6. Badilisha thamani ulizopewa.

    12(0.4771−1+0.3010)\frac{1}{2} (0.4771 - 1 + 0.3010)

  7. Fanya hesabu.

    12(0.7781−1)=12(−0.2219)=−0.11095\frac{1}{2} (0.7781 - 1) = \frac{1}{2} (-0.2219) = -0.11095

    Jibu la mwisho kwa b) ni:

    −0.1110\boxed{-0.1110}

    4 done, 1 left today. You're doing great.

Was this helpful?

Try one yourself

A new problem that tests the same skill.

Still stuck on this one?

Ask a follow-up, and the answer starts from this question.

Got a different question?Ask your own question

More Mathematics Questions

QGiven that log 10 2 = 0.3010 and log 10 3 = 0.4771, evaluate correct to 2 significant figures and…QGiven that log3 = 0.4771 and log2 = 0.3010, evaluate log162 + log36 - log54.QGiven that log 2 = 0.3010 and log 3 = 0.4771, find the value of log(1.5)".QGiven that 3 = 0.4771 and 2 = 0.3010. We assume the logarithm is base 10.QA) Show that log(70/33) + log(22/135) - log(7/18) = 3log2 - 2log3. b) If log10 x = a, compute the…
See all Mathematics questions →
Got a syllabus? Photograph it and I'll tell you what's due and what's worth the most.
Handwritten step-by-step solution preview

Handwritten Step-by-Step Solution

Get this as a handwritten image on WhatsApp — like a tutor wrote it for you.

Try on WhatsApp