Logarithms

log_b(a) is “log of a to the base b”.

If y = log_ax, then x = a^y

Properties

log(a) + log(b) = log(ab)

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

n log(a) = log(aⁿ)

log_b(a) = (base changing rule)

log_a(b) =

log_e(x) is also denoted by ln(x) and is called natural logarithm.

Note

Usually four place logarithm tables (base10) with antilog is available.

To find cube root of a number say 23,

let y = =

taking log to the base 10 (or any suitable base)

log(y) = = 0.4539092787

taking antilog we get

y = 10^0.4539092787

or y = 2.84386698

so that = 2.84386698

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some information on base 10 logarithm

log₁₀(2.35) = 0.3711

0 is called the characteristic and 0.3711 is called the mantissa

(hint: 23.5 = 2.35 X 10⁰)

log₁₀(23.5) = 1.3711

(hint: 23.5 = 2.35 X 10¹ )

1 is called the characteristic and 0.3711 is called the mantissa

log₁₀(235) = 2.3711

(hint: 235 = 2.35 X 10² )

log₁₀(2350) =3.3711

(hint: 2350 = 2.35 X 10³ )

log₁₀(23500) =4.3710

(hint: 23500 = 2.35 X 10⁴ )

log₁₀(235000) =5.3711

(hint: 235000 = 2.35 X 10⁵ )

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log₁₀(0.235) = or -0.6289

(hint: 0.235 = 2.35 X 10^(-1) )

log₁₀(0.0235) = or -1.6289

(hint: 0.0235 = 2.35 X 10^(-2) )

the bar above the characteristic shows that only the characteristic part is negative and the mantissa is positive

for example = (-1) + 0.3711 = -0.6289 as shown above

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