# Exponential equations

## Supporting facts

Any increasing (decreasing) function between each acquires its value only in one point of this period. When the exponential function is increasing.

When the exponential function comes in.

When the exponential function was.

The decision model equations one must know the properties of roots and degrees.

## Examples of solution of simple exponential equations

1. 2. Solution:   Answer: 3. 4. Solution:   Answer: 5. 6. Solution:

No roots (so )

7. 8. Solution:  Answer: ## Examples of the solution of model equations by reduction to the simplest

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If the left and right parts of the equations are illustrative only of the work, fractions, roots AO extent, it is advisable using basic formulas to try to record both parts of the equation as powers of one base.

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### Example 1:

Rozwarte equations .

Solutions:    Answer: .

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If one part of the exponential equation is a number and the other containing all the members of the expression (exponents differ only by free members), it is convenient in this part of the equation to put aside the smallest degree .

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### Example 2:

Rozwarte equations .

Solutions:    Answer: .

## Examples of more complex exponential equations

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Get rid of numeric terms in the exponents (using right-to-left basic properties of degrees).

If possible, reduce all degrees to one basis and perform a change of variables.

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### Example 3:

Rozwarte equations .

Solutions: Given that , we reduce the degree by one base 2: Substitution gives the equation: Backward substitution gives the equation , where or - no roots.

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If not degree can be reduced to one basis, try to reduce all degrees to two bases so as to obtain a homogeneous equation.

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### Example 4:

Rozwarte equations .

Solutions:

Let's give all the extent to the two bases 2 and 3: Have a homogeneous equation. For its solution divide both sides by ; Substitution gives the equation: Backward substitution gives the equation , where or - no roots.

Answer: ______________________________________________________________________

In other cases, we transfer all the terms in the equation into one part and try to decompose the obtained expression into factors, or apply special techniques to the solution in which we use properties of the corresponding functions.

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### Example 5:

Rozwarte equations .

Solutions:

If in pairs to group members in the left side of the equation, and in each pair stand out a common factor, we get : Make the brackets a common factor : Then or . We get two equations 1) where 2) where .

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