Question

㏒23∧-3 = ?
without using table or calculator

Answer 1

log23^-3

= -3log23

{logaⁿ = nloga}

20 < 23 < 25

Take logarithms for all,

log20 < log23 < log25

log 2×10 < log 23 < log 5×5

log 2 + log 10 < log 23 < log 5²

log 2 + 1 < log 23 < 2log5

We know that,

log 2 = 0.3031

log5 = 0.6989

Substitute these values:-

0.3031+1 < log 23 < 2(0.6989)

1.3031 < log 23 < 1.3978

Multiply with -3 so as to find log 23^-3

-3(1.3031) > -3log23 > -3(1.3978)

-3.9093 > -3log23 > -4.1934

-4.1934 < -3log23 < -3.9093

-4.1934 < log(23)^-3 < -3.9093

Now,

log 23^-3 = average of -4.1934 and -3.9093

log 23^-3 = (-4.1934+(-3.9093))/2

→ = (-4.1934-3.9093)/2

→ = -8.1027/2

→ = -4.05135

Approximately, log 23^-3 = -4

Hope it helps

Answer 2

Exact value we can't get without using table or calculator but we can find out approximately value by using some rule of Logarithms .

Here,

log(23)^-3

[ we know, logxⁿ = nlogx ]

= -3log(23)

we know,

20 < 23 < 25

log20 < log23 < log25

log(10 ×2) < log23 < log(5×5)

log10 + log2 < log23 < 2log5

1 + log2 < log23 < 2log5

we know,
log2 = 0.3031
log5 = 0.6989
use , this here,

1 + 0.3031 < log23 < 2(0.6989)

1.3031 < log23 < 1.3978

multiply with -3 , then inequality change

-3(1.3031) > -3log23 > -3(1.3978)

-4.1934 < -3log23 < -3.9093

-4.1934 < log(23)^-3 < -3.9093

hence, value of log(23)^-3 is average of (-4.1934) and (-3.9093)

log(23)^-3 ≈ ( -4.1934 -3.9093)/2

log(23)^-3 ≈ -4.05135

hence, approximately value of
log(23)^-3 = -4.0