# Exponential Equations

Algebra Rules for Exponential Functions:

If $\displaystyle n\in N$, then an is the product of n a’s.

For example, $\displaystyle {{2}^{4}}=2\times 2\times 2\times 2=16$

$\displaystyle {{a}^{0}}=1$

$\displaystyle \begin{array}{l}if\,\,n,\,\,m\,\in N,\,\,then\\{{a}^{{\frac{n}{m}}}}=\sqrt[m]{{{{a}^{n}}}}\,={{(\sqrt[m]{a}\,)}^{n}}\\{{a}^{{-x}}}=\frac{1}{{{{a}^{x}}}}\end{array}$

$\displaystyle \begin{array}{l}{{a}^{x}}{{a}^{y}}={{a}^{{a+y}}}\\\frac{{{{a}^{x}}}}{{{{a}^{y}}}}={{a}^{{x-y}}}\\{{({{a}^{x}})}^{y}}={{a}^{{xy}}}\end{array}$

Examples of exponential equations with the same bases

Example No.1:

Solve $\displaystyle {{2}^{{x+1}}}={{2}^{4}}$

Solution:

$\displaystyle {{2}^{{x+1}}}={{2}^{4}}$

$\displaystyle x+1=4$   (Equate the exponents)

$\displaystyle x+1-1=4-1$   (Subtract 1 from both sides)

$\displaystyle x=3$     (simplify it)

Example No.2:

Solve $\displaystyle 2={{2}^{{2x-4}}}$

Solution:

$\displaystyle 2={{2}^{{2x-4}}}$

$\displaystyle 1=2x-4$   (Equate the exponents)

$\displaystyle 1+4=2x-4+4$   (Add 4 from both sides)

$\displaystyle 5=2x$     (simplify it)

$\displaystyle \frac{5}{2}=\frac{{2x}}{2}$       (Divide on both sides by 2)

$\displaystyle \frac{5}{2}=x$      (finally simplify it)

Examples of exponential equations with the unlike bases

Example No.1:

Solve $\displaystyle {{4}^{x}}=64$

Solution:

$\displaystyle {{4}^{x}}=64$

$\displaystyle {{4}^{x}}={{4}^{3}}$     (Rewrite 64 as 43)

$\displaystyle x=3$     (Equate the exponent to obtain the value of x)

Example No.2:

Solve $\displaystyle {{9}^{x}}={{3}^{{x-2}}}$

Solution:

$\displaystyle {{9}^{x}}={{3}^{{x-2}}}$

$\displaystyle {{({{3}^{2}})}^{x}}={{3}^{{x-2}}}$    (Rewrite 9 as 32)

$\displaystyle {{3}^{{2x}}}={{3}^{{x-2}}}$     (Power of a power property)

$\displaystyle 2x=x-2$    (Equate the exponents)

$\displaystyle 2x-x=x-x-2$     (Subtract both side by 2)

$\displaystyle x=-2$    (Simplify it)

Examples of solving exponential equation when 0 < b < 1

Example No.1:

Solve  $\displaystyle {{\left( {\frac{1}{3}} \right)}^{x}}=9$

Solution:

$\displaystyle {{\left( {\frac{1}{3}} \right)}^{x}}=9$

$\displaystyle {{({{3}^{{-1}}})}^{x}}={{3}^{2}}$      (Rewrite 1/2 as 2-1 and 9 as 32)

$\displaystyle {{3}^{{-x}}}={{3}^{2}}$    (Power of a power property)

$\displaystyle -x=2$     (Equate the exponents)

$\displaystyle x=2$   (Solve for x)

Example No.2:

Solve  $\displaystyle {{2}^{{x+1}}}=\frac{1}{{16}}$

Solution:

$\displaystyle {{2}^{{x+1}}}=\frac{1}{{16} }$

$\displaystyle {{2}^{{x+1}}}=\frac{1}{{{{2}^{4}}}}$   (Rewrite 16 as 24)

$\displaystyle {{2}^{{x+1}}}={{2}^{{-4}}}$    (Definition of negative exponent)

$\displaystyle x+1=-4$     (Equate the exponents)

$\displaystyle x+1-1=-4-1$     (Subtract both sides by 1)

$\displaystyle x=-5$    (solve for x)

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