Solving Logarithmic Equations


Solving Logarithmic Equations


Recall the inverse nature of logs and exponential functions when solving logarithmic equations. 

y = bx    is the same as    logb(y) = x

Recall that natural logs have a base of e, and are written as y = ln x.  Where no base is specified a base 10 is assumed, written as y = log x.


Example

Solve:     log2(x) + log2(x - 2) = 3


Understand

Apply the log rules for same bases to combine the terms on the left hand side of the equation.  Once the equation is a log equals a number, convert the logarithm to exponential form.


Solution

log2(x) + log2(x - 2) = 3


log2((x)(x - 2)) = 3
log2(x2 - 2x) = 3
23 = x2 - 2x
8 = x2 - 2x
x2 - 2x - 8 = 0
(x - 4)(x + 2) = 0
x = 4, -2

Check Answers

Check to make sure that no answer choice creates a base or argument in the log equal to zero or a negative number.

x = -2 will give a negative number in the argument for the original equation log2(x)

x = 4 will not produce a negative argument

Therefore, the only solution is  x = 4