Logarithms
Definition: The logarithm of a number to a given base is the exponent to
which this base must be raised to yield the given number.
logb N = x is equivalent to bx = N.
For example, the equality 53 = 125 may be written log5 125 = 3.
For computational purposes we usually use 10 as the base; if we write
log n, the base 10 is understood. Logarithms to base 10 are called
common logarithms.
The function inverse to the function y = bx, b > 0, b ≠ 1 is y = logb x.
This function is defined only for positive values of x.
The laws of logarithms are derived from the laws of exponents. They are
listed below.
1. logb 1= 0, b > 0, b ≠ 1.
2. logb b = 1, b > 0, b ≠ 1.
3. logb x + logb y = logb (xy), x > 0, y > 0, b > 0, b ≠ 1.
𝑥
4. logb x – logb y = logb ( ), x > 0, y > 0, b > 0, b ≠ 1
𝑦
5. logb xk = k ∙ logb x, x > 0, b > 0, b ≠ 1
Although common logarithms are generally used for computation,
logarithms to base e are used in more advanced work, particularly in
calculus. The constant e = 2.7183 … is an irrational number and is
significant in the study of organic growth and decay. The function y = ex
is usually called the exponential function.
Illustrative Problems
Example Find the value of log4 64.
Solution log4 64 = 3, since 43 = 64.
Example Which of the following values of x that satisfy log 𝑥 64 = 2?
A2
B4
C6
D8
E 10
Solution (D)
log 𝑥 64 = 2, 𝑥 2 = 64, x = 8
Example Which of the following values of x that satisfy log 3 27 = 𝑥?
A2
B3
C4
D5
Solution (B)
log 3 27 = 𝑥, 3𝑥 = 27, x = 3
E6
Example If log 63.8 = 1.8048, what is log 6.38?
Solution
63.8
log 6.38 = log
= log 63.8 – log 10
10
= 1.8048 – 1
= 0.8048.
Example If log 2 = a and log 3 = b, express log 12 in terms of a and b.
Solution
log 12 = log (3 ∙ 22 ) = log 3 + log 22 = log 3 + 2log 2 = b + 2a.
Example In the formula A = P (1 + r)n, express n in terms of A, P, and r.
Solution
𝐴
𝐴
(1 + r) n = ⇒ log(1 + 𝑟)𝑛 = log
𝑃
𝑝
⇒ n log (1 + r) = log A – log p
⇒n=
log 𝐴 – log 𝑝
log (1 + 𝑟)
Example If log t2 = 0.8762, log (100t) = ?
Solution
log t2 = 0.8762 ⇒ 2 log t = 0.8762 ⇒ log t = 0.4381
log (100t) = log 100 + log t = 2 + 0.4381 = 2.4381.
Example If log tan x = 0, find the least positive value of x.
𝜋
Solution If log tan x = 0, then tan x = 1. Therefore, x = .
4
Example If loga 2 = x and loga 5 = y, express loga 40 in terms of x and y.
Solution
loga 40 = loga (23 ∙ 5) = loga 23 + loga 5 = 3loga 2 + loga 5 = 3x + y.
Example Find log3 3√3.
3
2
(A) 3 (B) 1 (C) (D)
(E) none of these
2
3
Solution (C)
3
2
3
log3 3√3 = log 3 3 = .
2
Example If log x = 1.5877 and log y = 2.8476, what is the numerical
value of log(𝑥 3√𝑦)?
Solution
log (𝑥 3√𝑦) = log 𝑥 + log 3√𝑦
1
= log x + log 𝑦
3
1
= 1.5877 + (2.8476) = 2.5369.
3
Example The expression logb x = 1 + c is equivalent to
(A) b1 + c = x
(B) x1 + c = b
(C) b + bc = x
(D) x = (1 + c)b
(E) b1 – x = c
Solution The correct answer is (A).
log 𝑏 𝑥 = 1 + c ⇒ b1 + c = x.
Example The expression log (2xy) is equivalent to
(A) 2(log x + log y)
(B) 2(log x) (log y)
(C) 2log x + log y
(D) log 2 + log x + log y
(E) log x + 2log y
Solution The correct answer is (D).
log (2xy) = log 2 + log x + log y
Example The diagram below represents the graph of which equation?
(A) y = 2x
(B) y = 102 (C) y = log 2 𝑥 (D) y = log x (E) y = 10 log x
Solution The correct answer is (C).
y = log 2 𝑥
Example log3 92 is between what pair of consecutive integers?
Solution
log3 81 < log3 92 < log3 243 ⇒ 4 < log3 92 < log3 5
⇒ x is between 4 and 5
Example If P = K∙ 10– xt, x equals
Solution (D)
𝐾
𝐾
P = K∙ 10– xt ⇒ 10 xt =
⇒ log (10𝑥𝑡 ) = log ( )
𝑃
⇒ 𝑥𝑡 = log 𝐾 − log 𝑃 ⇒ 𝑥 =
𝑃
log 𝐾 − log 𝑃
𝑡
𝑟
Example If logr 6 = S and logr 3 = T, log 𝑟 ( ) is equal to
2
1
(A) log 2 𝑟
2
(B) 1 – S + T
(C) 1 – S – T
(D) logr 2 – 1
(E) 1 + S + T
Solution (B)
𝑟
6
2
3
log 𝑟 ( ) = log 𝑟 𝑟 − log 𝑟 2 = 1 − log 𝑟
= 1 − (log 𝑟 6 − log 𝑟 3)
= 1 − (𝑆 − 𝑇)
= 1 − 𝑆 + 𝑇.