Solving Logarithmic Equations

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

y = b^x    is the same as    log_b(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