Solving Exponential Equations

Solving Exponential Equations

Set 1: With a Common Base

  1. Simplify each side of the equation using properties of exponents.
  2. Rewrite the equation so both sides have the same base.
  3. Drop the bases and set the exponents equal to each other.
  • Example:
    2x+1=292^{x+1} = 2^9
    x+1=9x + 1 = 9
    x=8x = 8

Set 2: Without a Common Base

  • Rewrite each side of the equation using a common base.

  • Follow the same steps as in Set 1.

  • Example:
    62x−10=366^{2x-10} = 36
    62x−10=626^{2x-10} = 6^2
    2x−10=22x - 10 = 2
    2x=122x = 12
    x=6x = 6

Examples with Solutions

  • 54n+5=5n−75^{4n+5} = 5^{n-7}
    4n+5=n−74n + 5 = n - 7
    3n=−123n = -12
    n=−4n = -4

  • 3⋅3k+2=35k−13 \cdot 3^{k+2} = 3^{5k-1}
    3k+3=35k−13^{k+3} = 3^{5k-1}
    k+3=5k−1k + 3 = 5k - 1
    4=4k4 = 4k
    k=1k = 1

  • 10⋅102v−11=10−4⋅10910 \cdot 10^{2v-11} = 10^{-4} \cdot 10^9
    102v−10=10510^{2v-10} = 10^5
    2v−10=52v - 10 = 5
    2v=152v = 15
    v=152v = \frac{15}{2}

  • 2p−7=82^{p-7} = 8
    2p−7=232^{p-7} = 2^3
    p−7=3p - 7 = 3
    p=10p = 10

  • 32=22m−93^2 = 2^{2m-9}
    35=22m−93^5 = 2^{2m-9}
    243=22m−9243 = 2^{2m-9}

  • 4y+2=16y−34^{y+2} = 16^{y-3}
    4y+2=(42)y−34^{y+2} = (4^2)^{y-3}
    y+2=2(y−3)y + 2 = 2(y-3)
    y+2=2y−6y + 2 = 2y - 6
    8=y8 = y

  • 125y=25125^y = 25
    (53)y=52(5^3)^y = 5^2
    3y=23y = 2
    y=23y = \frac{2}{3}

  • 163x=8x+216^{3x} = 8^{x+2}
    (24)3x=(23)x+2(2^4)^{3x} = (2^3)^{x+2}
    12x=3x+612x = 3x + 6
    9x=69x = 6
    x=23x = \frac{2}{3}

  • 43x=8x−14^{3x} = 8^{x-1}
    (22)3x=(23)x−1(2^2)^{3x} = (2^3)^{x-1}
    6x=3x−36x = 3x - 3
    3x=−33x = -3
    x=−1x = -1

  • 82a−1=322a+18^{2a-1} = 32^{2a+1}
    (23)2a−1=(25)2a+1(2^3)^{2a-1} = (2^5)^{2a+1}
    6a−3=10a+56a - 3 = 10a + 5
    −8=4a-8 = 4a
    a=−2a = -2

  • 812x+54=(13)2x81^{\frac{2x+5}{4}} = (\frac{1}{3})^{2x}
    (34)2x+54=(3−1)2x(3^4)^{\frac{2x+5}{4}} = (3^{-1})^{2x}
    8x+20=−2x8x + 20 = -2x
    20=−10x20 = -10x
    x=−2x = -2

  • 272x=243x−227^{2x} = 243^{x-2}
    (33)2x=(35)x−2(3^3)^{2x} = (3^5)^{x-2}
    6x=5x−106x = 5x - 10
    x=−10x = -10

  • 64=4⋅44x64 = 4 \cdot 4^{4x}
    43=41⋅44x4^3 = 4^1 \cdot 4^{4x}
    3=1+4x3 = 1 + 4x
    2=4x2 = 4x
    x=12x = \frac{1}{2}

  • 92x+4⋅92x=1819^{2x+4} \cdot 9^{2x} = \frac{1}{81}
    92x+4⋅92x=9−29^{2x+4} \cdot 9^{2x} = 9^{-2}
    2x+4+2x=−22x + 4 + 2x = -2
    4x+4=−24x + 4 = -2
    4x=−64x = -6
    x=−32x = -\frac{3}{2}

  • 49x−5⋅7x−9=1749^{x-5} \cdot 7^{x-9} = \frac{1}{7}
    (72)x−5⋅7x−9=7−1(7^2)^{x-5} \cdot 7^{x-9} = 7^{-1}
    2x−10+x−9=−12x - 10 + x - 9 = -1
    3x−19=−13x - 19 = -1
    3x=183x = 18
    x=6x = 6

  • 42x16=46x+18\frac{4^{2x}}{16} = 4^{6x+18}
    42x42=46x+18\frac{4^{2x}}{4^2} = 4^{6x+18}
    42x−2=46x+184^{2x-2} = 4^{6x+18}
    2x−2=6x+182x - 2 = 6x + 18
    −20=4x-20 = 4x
    x=−5x = -5