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Logarithm Equation Examples

Logarithm Equation Example 1

Question:
If log1625 = xlog25 then x equals ?
Solution
Let y = log1625 = xlog25
when y = log1625
then 16y = 25
then (24)y = 52
then (2y)4 = 52 ...equation (1)
when y = xlog25
then y = log25x
then 2y = 5x ...equation (2)
Substitute the expression for 2y in equation (2) into equation (1)
(5x)4 = 52
then 54x = 52 then 4x = 2 then x = 1/2

Logarithm Equation Example 2

Question:
log216-0.5 = ?
Solution
log216-0.5 = - 0.5 log216 = - 0.5 log224 = - 0.5 × 4 log22 = - 0.5 × 4 = - 2
formulas used
(1). logabx = x logab
(2). logaa = 1

Logarithm Equation Example 3

Question:
What is the domain of the function log(2x2 - 3) ?
Solution
The domain of a log(x) function is that the variable x must be greater than zero.
From this, we have 2x2 - 3 > 0
2x2 > 3
x2 > 1.5
then x < -1.22 or x > 1.22
Therefore, the domain of the given function is x must be larger than 1.22 or smaller than -1.22

Exponential Equation Example 1

Question:
Solving the equation xlg x = 1000 x2
Solution:
From the definition of logarithm, if ax = b (a > 0, and a is not 0) then x = logab
If a = 10, then log10b, marked as lg b,
Take the lg of both side ( note: lg b is log10b)
lg(xlg x) = lg(1000 x2)
(lg x)(lg x) = lg1000 + lg x2
(lg x)2 = lg 103 + 2 lg x
(lg x)2 - 2 lg x - 3 = 0
Factoring this quadratic equation, (lg x + 1)(lg x - 3) = 0
lg x + 1 = 0 or lg x - 3 = 0
For lg x + 1 = 0 => lg x = -1 => 10-1 = x => x = 1/10
For lg x - 3 = 0 => lg x = 3 => 103 = x => x = 1000
formulas used:
logabx = x logab
logaa = 1
loga(MN) = logaM + logaN

Exponential Equation Example 2

Question:
f(x) = 2x + 3 is a function. If f(x) = 10, find x = ?
Solution
Given f(x) = 10, then 2x + 3 = 10 => 2x = 7
Take log to both sides
log2x = log 7 => x log2 = log 7
x = log 7/log 2 = 0.8451/0.3010 = 2.8076