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}

Exponential Equations Examples
Exponential Equations Examples

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)