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Wize High School Grade 12 Pre-Calculus Textbook > Exponential Functions

Solving Exponential Equations

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Solving Exponential Equations

An exponential equation is an equation where the variable is in the exponent.

How to Solve an Exponential Equation

  • Write both sides of the equation in the same base.
  • Equate the exponents on both sides of the equation.
  • Solver for the value of the variable.

Example

Solve for x: 7x−2=3437^{x-2}=343


7x−2=3437x−2=73x−2=3x=5\begin{array}{rcl} 7^{x-2}&=&343\\\\ 7^{x-2}&=&7^3\\\\ x-2&=&3\\\\ x&=&5 \end{array}
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Example: Solving Exponential Equations

Solve for x: 35x−1=813x3^{5x-1}=81^{3x}


35x−1=813x35x−1=(34)3x35x−1=312x5x−1=12x−7x=1x=−17\begin{array}{rcl} 3^{5x-1}&=&81^{3x}\\\\ 3^{5x-1}&=&(3^{4})^{3x}\\\\ 3^{5x-1}&=&3^{12x}\\\\ 5x-1&=&12x\\\\ -7x&=&1\\\\ x&=&-\dfrac{1}{7} \end{array}

Practice: Solving Exponential Equations

Solve for x: 93x+1=273x9^{3x+1}=27^{3x}. Leave answer as a fraction.

Practice: Solving Exponential Equations

What value of xx makes the following statement true?

(47)5x=(64343)2x−1\Bigg(\dfrac{4}{7}\Bigg)^{5x}=\Bigg(\dfrac{64}{343}\Bigg)^{2x-1}

Practice: Solving Exponential Equations

Solve for x: 2(62x)−74(6x)+72=02(6^{2x})-74(6^x)+72=0
Extra Practice
Solving Exponential Equations
Solve 5(1128)2x+5=10(32)x+15\Bigg(\dfrac{1}{128}\Bigg)^{2x+5}=10(32)^{x+1}
Solving Exponential Equations
Solve 4(8)3x+1=32(4)x+24(8)^{3x+1}=32(4)^{x+2}
Solving Exponential Equations
Practice: Solving Exponential Equations
Solve for x: 4x−6(2x)+8=04^x-6(2^x)+8=0
Solving Exponential Equations
Practice: Solving Exponential Equations
Solve for x: 32x−3x=03^{2x}-3^x=0
Solving Exponential Equations
Practice: Solving Exponential Equations
What value of xx makes the following statement true? Leave answer in exact form.
(216125)3x=(3625)2x−1\Bigg(\dfrac{216}{125}\Bigg)^{3x}=\Bigg(\dfrac{36}{25}\Bigg)^{2x-1}
Solving Exponential Equations
Practice: Solving Exponential Equations
What value of xx makes the following statement true? Leave answer in exact form.
(34)2x−1=(81256)x+2\Bigg(\dfrac{3}{4}\Bigg)^{2x-1}=\Bigg(\dfrac{81}{256}\Bigg)^{x+2}
Solving Exponential Equations
Practice: Solving Exponential Equations
Solve for x: 254x+5=125x−325^{4x+5}=125^{x-3}. Leave answer as a fraction.
Solving Exponential Equations
Practice: Solving Exponential Equations
Solve for x: 42x+1=83x−14^{2x+1}=8^{3x-1}. Leave answer as a fraction.