Solving Logarithmic Equations

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

y = b^x    is the same as    log_b(y) = x

Where no base is specified, as in y = log x, a base 10 is assumed. So log x = log₁₀ x.  This is known as a common logarithm.

A natural logarithm is one that has a base of e, an irrational number approximately equal to 2.71828.  The natural log is written as y = ln x. 

Like with exponential functions, logarithmic functions are one-to-one.  So if you have a single logarithm on either side of an equation, you can set the inputs (arguments) equal to each other and solve, as long as the bases are equal.

log₅(x + 3) = log₅ 12
x + 3 = 12
x = 9
 

Properties of Logarithms 

When solving logarithmic equations, consider the following properties of logarithms.  Note that these properties apply to logs with the same base.

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

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.