Question

30) logs 32 Use a calculator to evaluate the logarithm. 31) log 2.32 32) In 111 32) Write the expression as the logarithm of a single number or expression with a coefficient of 1. Assume all variables represent positive numbers. 33) log 6 + log 54 - log 9 33) 34) 3 In (x + 3) - 6 In (x2 + 7) 34)

Answer 1

1. Log₈(32) = Log₈(4x8)

log(ab) = log a + log b

Log₈(32) = Log₈(4) + Log₈(8)

= 2*Log₈(2) + Log₈(8)

log_nn = 1 ( n> 0)

so, log₈8 = 1

also, log_ab = 1/ log_ba

Log₈(2) = 1/Log₂(8) = 1/ Log₂(2^3) = 1/3

So, the value of expression = 1/3 + 1 = 4/3

2. Log(2.32)

Characteristic of 2.32x 10^0 = 0 (exponent of 10)

Mantissa(using log table)

Select the row with 23(first two terms)

Select the column with 2(third number)

There intersection is 3655

Also, there's no non zero term after third term which means, there will be no addition to mantissa.

So, log(2.32) = 0.3655

3. ln 111

we can calculate log (base 10) and then convert it to ln

ln = 2.303* log(base 10)

let's find log(111)

char of log 111 is power of 10 in 1.11*10^2 => 2

mantissa:

choose row from 11 (first two terms)

choose column from 1 (third term)

their intersection = 0453

as fourth term is not non-zero. no need to add anything to mantissa.

log(111) = 2.0453

ln 111 = 2.303*2.0453 = 4.710

4. log 6 + log 54 - log 9

using property of log

log a + log b = log(ab)

log a - log b = log(a/b)

above expression can be written as

log(6*54/9)

= log 36

characteristic of 3.6x10^1 = 1

mantissa:

choose row from 36

choose column from 0

intersection = 5563

log(36) = 1.5563