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Exponential equations are equations where the variable we are looking to unravel appears in the exponent.

If an exponential equation has the same base on each side of the equation, then the exponent is equal. 

๐Ÿ‘‰This post extends our series on indices๐Ÿ‘ˆ

Example 1
If $4^{x-1}=4^{2x-4}$, find the value of $x$

The base($4$) is the same, hence, the exponents are equal

Collecting like term

Divide both sides by $-1$

Example 2
Solve for x in the equation

Unlike the previous example, the common base is not explicitly shown. To solve this,  we will rewrite 81 as a base of 3


The base is the same, hence

Collecting like terms

Divide both sides by 3

Example 3
Solve the exponential equation

$5$ is the same as $5^1$, hence

You know that $\frac{a^3}{a^2}=a^{3-2}$, hence

The base is the samesidesis,

Divide through by 2

Example 4
Solve for x in the equation

$\sqrt[3]{81}$ can be rewritten as $3^{\frac{4}{3}}$, hence,

The base on both sides are the same, thus, we equate the exponent

Example 5
Solve $\sqrt[5]{\frac{1}{243}}=3^x$

First, we express $\frac{1}{243}$ as a base of 3. Accordingly  $\frac{1}{243}=3^{-5}$, then


This translates into:


Example 6
Solve $\frac{1}{81^{(x-2)}}=27^{(1-x)}$

Rewritting to the power of $3$

Equating the power

Collecting like terms

Divide both sides by $-1$

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Example 7
Solve for x if

Expressing the right hand side to the base of 2

Simplifying the indices

Equating the exponent

Taking the L.C.M

Divide both sides by $8$

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