Exponential and Logarithmic Equations

The equations provided are meant to illustrate techniques for solving exponential and logarithmic equations. Here’s a breakdown for better understanding:

Exponential Equations

  1. Find the solution set of the exponential equation: a) 23x−8=162^{3x-8} = 16

    • Rewrite 1616 as a power of 22: 23x−8=242^{3x-8} = 2^4.

    • Set the exponents equal: 3x−8=43x - 8 = 4.

    • Solve for xx: 3x=123x = 12 x=4x = 4. b) 27x+3=9x−127^{x+3} = 9^{x-1}

      • Rewrite as powers of 33: 33(x+3)=32(x−1)3^{3(x+3)} = 3^{2(x-1)}.

      • Set exponents equal: 3(x+3)=2(x−1)3(x + 3) = 2(x - 1).

      • Solve for xx.
        c) 32−x=rac1273^{2-x} = rac{1}{27}

    • Recognize rac127=3−3rac{1}{27} = 3^{-3}, then 32−x=3−33^{2-x} = 3^{-3} leads to 2−x=−32 - x = -3. d) 0.15x=0.01−3x0.15^x = 0.01^{-3x}

      • Rewrite 0.010.01 as 0.15−20.15^{-2}: 0.15x=(0.15−2)−3x0.15^x = (0.15^{-2})^{-3x}.

Logarithms for Solving Exponential Equations

  1. Using logs to isolate the exponent: If you have an equation where you can't easily rewrite the bases, use natural or common logarithms: a) 4x=154^x = 15

    • Take extlnext{ln} of both sides: xextln(4)=extln(15)x ext{ln}(4) = ext{ln}(15), then solve for xx.

Logarithmic Equations

  1. Express logarithmic equations in exponential form: Example: extlogbM=cext{log}_b M = c converts to bc=Mb^c = M. a) 3extln(2x)=123 ext{ln}(2x) = 12

    • Convert to extln(2x)=4ext{ln}(2x) = 4 and solve as explained.