Logarithm- QA Formulae

By BYJU'S CAT|Updated : June 8th, 2023

Logarithm is the exponent or power to which a base must be raised to yield a given number. x is the logarithm of n to the base a if a^x = n, in which case one writes x = log_an, when expressed mathematically.

For example, 2³ = 8; therefore, 3 is the logarithm of 8 to base 2, or 3 = log₂ 8. Since 10² = 100, 2 = log₁₀ 100 in the same fashion.

Another way to understand this would be:

If a^x = N, then x = log_an

Table of Content

1. Properties of logarithm:
2. SOME POINTS TO REMEMBER:
3. Log values of some numbers (base 10)

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Logarithm of a negative number or zero is not defined.

Natural logarithm: Base of the number is ‘e’.

Common logarithm: Base of the number is 10. When the base is not mentioned, it can be taken as 10.

Properties of logarithm:

log_a1 = 0
log_aa = 1
log_axy = log_ax + log_ay
log_a(X/Y) = log_aX – log_aY
log_ab × log_ba = 1
_b^logbx = x
log_ax = 1/log_xa
log_ab = log_cb × log_ac
log_ab^c = c × log_ab
xlog_by = ylog_bx
If 0 < a < 1, then log_ax < log_ay (if x > y)
If a > 1, then log_ax > log_ay (if x > y)

SOME POINTS TO REMEMBER:

log(x – y) ≠ logx – logy
log(x + y) ≠ logx + logy

Log values of some numbers (base 10)

Number	Value
1	0
2	0.301
3	0.4771
4	0.602
5	0.698
6	0.778
7	0.845
8	0.903
9	0.954

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