Solving exponential equations with different bases without logarithms

Exponential Equations

Content of this page:

• Introduction

• Properties of Powers

• 25 Resolved Exponential Equations step by step

Introduction

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

In this section, we will resolve the 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.

For example:

$$ 3^{2x} = 3^6 $$

Obviously, the value that x has to take for the equality to be true is 3.

In order to achieve this type of expressions we have to factorize, express all the numbers in the form of powers, apply the properties of powers and write roots as powers.

Sometimes we will need to make a change of variable to transform the equation in a quadratic one.

We can also resolve using logarithms, but we will leave this type of procedures for more difficult equations with different bases in the exponential expressions, making it impossible to use the previous method of equalizing.

For example,

$$ 3^{x+3} = 5^x $$

which has a real solution, using logarithms of,

$$ x = \frac{3 ln(3)}{ln\left(\frac{5}{3}\right)} $$

Before we start...let's remember the properties of powers

Product	Power
Quotient	Negative exponent
Inverse	Inverse of inverse

Solved Exponential Equations

Equation 1

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Equation 2

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Equation 3

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Equation 4

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Equation 5

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Equation 6

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Equation 7

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Equation 8

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Equation 9

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Equation 10

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Equation 11

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Equation 12

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Equation 13

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Equation 14

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Equation 15

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Equation 16

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Equation 17

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Equation 18

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Equation 19

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Equation 20

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Equation 21

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Equation 22

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Equation 23

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Equation 24

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Equation 25

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How do you solve exponential equations without common bases?