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Solving Exponential Equations

The simplest sort of exponential equation is of the formBy inspectionbecause However not all equations can be solved by inspection even if they are of the simple form given above.

The equationhas no simple integer solutions. This equation can only be solving by taking the log of both sides:

and then using the power rule for logs:

to 4 significant figures.

Slightly more complicated are expressions whereis a power on both sides, for example,

We log both sides to get

Rearrangement of this expression gives

Now we can factorise this expression with x as a common factor on the left hand side to give

to 4 decimal places.

Sometimes an equation may have to be rearranged to express it in one of the above forms. For example,We use the rule

Then we can log both sides as above

As before we rearrange this equation to make x the subject:

to 3 significant figures.