Exponential Equations

An exponential equation is one that has exponential expressions, in other words, powers that have in their exponent expressions with the unknown factor x. For example:

We will resolve some exponential equations without using logarithms. This method of resolution consists in reaching an equality of the exponentials with the same base in order to equal the exponents.

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Examples

Example 1:

Taking into account that we can write 16 as a power

we can rewrite the equation as

Therefore,

So, the solution is x = 2.

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Example 2:

Taking into account that

We can rewrite the equation as

This way, we can extract a common factor of 2 ^x:

Therefore, the solution is

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Example 3:

Taking into account that

We can rewrite the equation as

We have the common base 3 ^x, but because one of them is squared, we write

Substituting, the equation finishes like

In other words, a quadratic equation:

We multiply the full equation by 9:

We solve it:

Therefore,

So, we obtain

The second option is not possible because it is negative. Therefore,

From where we obtain

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More examples: 25 solved exponential equations

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