Question

Solve all 3 Questions. 50 Points + Brainelist ​

Answer 1

i) Using log law:
`\log_aa=1`

`\implies \log_55+1=1+1=2`

ii)
`\log \left((15)/(8)\right)+4 \log 2-\log 3`

Using log law
`a \log b=\log b^a`:

`\implies \log \left((15)/(8)\right)+\log 2^4-\log 3`

`\implies \log \left((15)/(8)\right)+\log 16-\log 3`

Using log law
`\log a-\log b=\log ((a)/(b))`:

`\implies \log \left((15)/(8)\right)+\log\left((16)/(3)\right)`

Using log law
`\log a+\log b=\log(ab)`:

`\implies \log \left((15)/(8)\cdot (16)/(3)\right)`

`\implies \log 10`

Using log law:
`\log_aa=1`

`\implies \log_(10) 10=1`

iii) Take log of base 10:

`\log_(10)(√(8.357)*0.895^2)`

`\implies \frac12\log_(10)(8.357)+2\log_(10)(0.895)`

Log tables

The characteristic of the logarithm of a number is the exponent of 10 in its scientific notation.

The mantissa is found using the log tables and is always prefixed by a decimal point.

The row is the first two non-zero digits of the number, and the column is the 3rd digit of the number

Use the log tables to find
`\log_(10)(8.357)`:

8.357 = 8.357 × 10⁰

⇒ characteristic = 0

log table: row 83, column 5 ⇒ mantissa 9217

(as there is a 4th digit) Mean difference 7 = 4

mantissa + mean difference = 9217 + 4 = 9221 ⇒ 0.9221

characteristic + mantissa = 0 + 0.9221 = 0.9221

Therefore,
`\log_(10)(8.357)=0.9221`

Use the log tables to find
`\log_(10)(0.895)`:

`0.895 = 8.95* 10^(-1)`

⇒ characteristic = -1

log table: row 89, column 5 ⇒ mantissa 9518⇒ 0.9518

characteristic + mantissa = -1 + 0.9518= -0.0482

Therefore,
`\log_(10)(0.895)=-0.0482`

Therefore,

`\implies \frac12\log_(10)(8.357)+2\log_(10)(0.895)`

`\implies \frac12 \cdot 0.9221+2\cdot-0.0482`

`\implies 0.36465`

Therefore,

`\log_(10)(√(8.357)*0.895^2)=0.36465`

Using
`\log_ab=c \implies a^c=b`

`\implies √(8.357)*0.895^2=10^(0.36465)`

`\implies √(8.357)*0.895^2=2.3155`