How do you solve log of x equals 2 for x?

Answer 1

The first type of logarithmic equation has two logs, each having the same base, set equal to each other, and you solve by setting the insides (the "arguments") equal to each other.

For example:

• Solve log2(x) = log2(14).

Since the logarithms on either side of the equation have the same base ("2", in this case), then the only way these two logs can be equal is for their arguments to be equal. In other words, the log expressions being equal says that the arguments must be equal, so I have:

x = 14

And that's the solution: x = 14

• Solve logb(x2) = logb(2x - 1).

ADVERTISEMENT

Since the bases of the logs are the same (the unknown value "b", in this case), then the insides must be equal. That is:

x2 = 2x - 1

Then I can solve the log equation by solving this quadratic equation:

x2 - 2x + 1 = 0

(x - 1)(x - 1) = 0

Then the solution is x = 1.

Logarithms cannot have non-positive arguments, but quadratics and other equations can have negative solutions. So it is generally a good idea to check the solutions you get for log equations:

logb(x2) = logb(2x - 1)

logb([1]2) ?=? logb(2[1] - 1)

logb(1) ?=? logb(2 - 1)

logb(1) = logb(1)

The value of the base of the log is irrelevant here. Each log has the same base, each log ends up with the same argument, and that argument is a positive value, so the solution "checks".