Definition 13.3.1. Exponential Equation.
An exponential equation is an equation with a variable in the exponent.
Section 13.3 Exponential Equations
Subsection 13.3.1 Solving Exponential Equations Using Common Bases
Some simple exponential equations can be solved by using the intuitive fact that if two powers with the same base are equal, then their exponents must be equal. In other words,
\begin{gather*}
\boxed{\text{If $b^x=b^y$, then $x=y$}}
\end{gather*}
This means that if we can write both sides of an exponential equation as a power of the same base, then we can set their exponents equal (equate exponents), resulting in a simpler equation that we can solve.
-
Rewrite both sides using a common base.
-
Simplify using exponent laws.
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Set the exponents equal.
-
Solve the resulting equation.
The resulting equation could be linear, quadratic, or some other type of equation.
Sometimes, both sides can be written as a power of multiple numbers, which result in the same solution.
Subsection 13.3.2 Examples
Exercise Group 13.3.1. Basic Examples.
Solve each exponential equation.
(a)
\(2^{x-2} = 64\)Answer.
\(x=8\)
(b)
\(5^x=625\)Answer.
\(x=4\)
(c)
\(2^{2x+2}+7=71\)Answer.
\(x=2\)
(d)
\(81=3^{x+1}\)Answer.
\(x=3\)
(e)
\(3^{x+1}=9^{x-1}\)Answer.
\(x=3\)
(f)
\(2^x=256\)Answer.
\(x=8\)
(g)
\(25^{x-1}=5^{3x}\)Answer.
\(x=-2\)
(h)
\(3^x=9^{x-2}\)Answer.
\(x=4\)
(i)
\(2^x = 8^{x-1}\)Answer.
\(x=\frac{3}{2}\)
(j)
\(4^{x-1}=2^{x+3}\)Answer.
\(x=5\)
(k)
\(27^x=9^{x-2}\)Answer.
\(x=-4\)
(l)
\(2^{5x+3}-64=0\)Answer.
\(x=\frac{3}{5}\)
(m)
\(4^{2x}=8^{x+1}\)Answer.
\(x=3\)
(n)
\(2^{4x}=4^{x+3}\)Answer.
\(x=3\)
(o)
\(4^{3x}=8^{x-3}\)Answer.
\(x=-3\)
(p)
\(4^{x+2}=64^x\)Answer.
\(x=1\)
(q)
\(16^x = 4\)Answer.
\(x=\frac{1}{2}\)
(r)
\(4^x = 8^{x-1}\)Answer.
\(x=3\)
(s)
\(3^{x-3} = 81\)Answer.
\(x=7\)
(t)
\(6^{x-3} = 1\)Answer.
\(x=3\)
(u)
\(64^{2x} = 16^{x+1}\)Answer.
\(x=\frac{1}{2}\)
(v)
\(9^{4x-2} = 81\)Answer.
\(x=1\)
(w)
\(9^{4x-9} = 729\)Answer.
\(x=3\)
Exercise Group 13.3.2. Intermediate Examples.
Solve each equation.
(a)
\(9^{x-1} = 27^{x+3}\)Answer.
\(x=-11\)
(b)
\(4^{2x}=2^{5-x}\)Answer.
\(x=1\)
(c)
\(2^x = \frac{1}{128}\)Answer.
\(x=-7\)
(d)
\(27^{x-4}=9^{x+3}\)Answer.
\(x=18\)
(e)
\(4 (9^{x-1}) = 108\)Answer.
\(x=\frac{5}{2}\)
(f)
\(8^{2x}=16^{x+3}\)Answer.
\(x=6\)
(g)
\(4^{2x}=8^{2x-3}\)Answer.
\(x=\frac{9}{2}\)
(h)
\(4^{2x+1}=8^{2x-1}\)Answer.
\(x=\frac{5}{2}\)
(i)
\(2^{3x}=\frac{1}{2}\)Answer.
\(x=-\frac{1}{3}\)
(j)
\(36^{3x} = 216^{x+7}\)Answer.
\(x = 7\)
(k)
\(4^{2x}=\frac{1}{16}\)Answer.
\(x=-1\)
(l)
\(36^{3x-1}=6^{2x+5}\)Answer.
\(x=\frac{7}{4}\)
(m)
\(16^{2x-3}=32^{x+3}\)Answer.
\(x=9\)
(n)
\(9^{3x+6} = 27^{x-3}\)Answer.
\(x=-7\)
(o)
\(\brac{\frac{1}{7}}^x = 343\)Answer.
\(x=-3\)
(p)
\(9^{x+1}=243^{x+3}\)Answer.
\(x=-\frac{13}{3}\)
(q)
\(125^{2x-1}=25^{x+4}\)Answer.
\(x=\frac{11}{4}\)
(r)
\(25^{x+7} = 625^{x-4}\)Answer.
\(x=15\)
(s)
\(4^{-2x+1}=8^{x-4}\)Answer.
\(x=2\)
(t)
\(5^{3x-6} = 125\)Answer.
\(x=3\)
(u)
\(9^{2x-3}=27^{1-x}\)Answer.
\(x=\frac{9}{7}\)
(v)
\(9^{2x+1}=81\cdot 27^x\)Answer.
\(x=2\)
(w)
\(3^{5x}=27^{x-1}\)Answer.
\(x=-\frac{3}{2}\)
Exercise Group 13.3.3. Equations with More Fractions and Radicals.
Solve each equation.
(a)
\((\sqrt{3})^{x+1} = 27^x\)Answer.
\(x=\frac{1}{5}\)
(b)
\(2^x=8\sqrt[3]{2}\)Answer.
\(x=\frac{10}{3}\)
(c)
\(\brac{\frac{5}{2}}^x = \frac{8}{125}\)Answer.
\(x=-3\)
(d)
\(8^{x-1}=\sqrt[3]{16}\)Answer.
\(x=\frac{13}{9}\)
(e)
\(81\sqrt{3}=3^x\)Answer.
\(x=\frac{9}{2}\)
(f)
\(20^x = 0.05\)Answer.
\(x=-1\)
(g)
\(2^{x+1}=2\sqrt[3]{4}\)Answer.
\(x=\frac{2}{3}\)
(h)
\(9^{x+1}=27^{2x-3}\)Answer.
\(x=\frac{11}{4}\)
(i)
\(9^x=\sqrt{27}\)Answer.
\(x=\frac{3}{4}\)
(j)
\(3^{2x}=9^{\frac{1}{2}(x-4)}\)Answer.
\(x=-4\)
(k)
\(\brac{\sqrt{7}}^{x+1}=\sqrt[3]{49}\)Answer.
\(x=\frac{1}{3}\)
(l)
\(\brac{\frac{1}{4}}^3=2^x\)Answer.
\(x=-6\)
(m)
\(5^x=\frac{\sqrt[3]{25}}{25}\)Answer.
\(x=-\frac{4}{3}\)
(n)
\(\frac{\sqrt[3]{49}}{343}=7^{x+1}\)Answer.
\(x=-\frac{10}{3}\)
(o)
\(\brac{\frac{1}{9}}^x=3\sqrt{27}\)Answer.
\(x=-\frac{5}{4}\)
(p)
\(49^{x-1}=7\sqrt{7}\)Answer.
\(x=\frac{7}{4}\)
(q)
\(\brac{\frac{1}{4}}^{x+4}=\sqrt{8}\)Answer.
\(x=-\frac{19}{4}\)
(r)
\(2^{3x-4}=0.25\)Answer.
\(x=\frac{2}{3}\)
(s)
\(36^{2x+4}=\brac{\sqrt{1296}}^x\)Answer.
\(x=-4\)
Exercise Group 13.3.4. Additional Practice.
Solve each equation.
(a)
\(\brac{\frac{1}{243}}^{4-x} = 3^{6x+3}\)Answer.
\(x=-23\)
(b)
\(7^{3-5x} = \brac{\frac{1}{49}}^{2x+9}\)Answer.
\(x=21\)
(c)
\(\brac{\frac{1}{8}}^{x+1}=\brac{\sqrt[3]{16}}^x\)Answer.
\(x=-\frac{9}{13}\)
(d)
\(125^{x+2} = \brac{\frac{1}{5}}^{1-5x}\)Answer.
\(x = \frac{7}{2}\)
(e)
\(2^{3-x} \cdot 4^{2x-1} = \frac{1}{16}\)Answer.
\(x=-\frac{5}{3}\)
(f)
\(1024^{2x-1}=16^{x+4}\)Answer.
\(x=\frac{13}{8}\)
(g)
\(\brac{\frac{1}{9}}^{2x-1}=27^{2-x}\)Answer.
\(x=-4\)
(h)
\(8^{1-x}=\frac{\sqrt[3]{16}}{4}\)Answer.
\(x=\frac{11}{9}\)
(i)
\(\sqrt[4]{216}=36^{x-1}\)Answer.
\(x=\frac{11}{8}\)
(j)
\(27^x = 81^{3x-1}\)Answer.
\(x=\frac{4}{9}\)
Exercise Group 13.3.5. Quadratic Exponents.
Solve each equation.
(a)
\(2^{x^2}=16\)Answer.
\(x=-2, 2\)
(b)
\(9^{2x-1}=\brac{\frac{1}{27}}^{x+2}\)Answer.
\(x=-\frac{4}{7}\)
(c)
\(9^{x+4}=3^{x^2}\)Answer.
\(x=-2, 44\)
(d)
\(4^{x^2-2x}=8^{1-x}\)Answer.
\(x=-1, \frac{3}{2}\)
(e)
\(\brac{\frac{1}{8}}^{2x+1}=32^{x-3}\)Answer.
\(x=\frac{12}{11}\)
(f)
\(4^{x^2-x}=1\)Answer.
\(x=0, 1\)
(g)
\(3^{x^2}=9\cdot 3^{-x}\)Answer.
\(x=-2,1\)
(h)
\(8^{2x-1} = \brac{\frac{1}{4}}^x (32)\)Answer.
\(x=1\)
(i)
\(2^{x^2}=32\brac{2^{4x}}\)Answer.
\(x=-1,5\)
(j)
\(3^{x^2+20}=\brac{\frac{1}{27}}^{3x}\)Answer.
\(x=-5,-4\)
Subsection 13.3.3 Advanced Examples
Checkpoint 13.3.2. Parameter Value Problem.
Exercise Group 13.3.6. Equations Reducible to Quadratics.
Solve each equation.
(a)
\(4^x+4^{x+1}=160\)Answer.
\(x = \frac{5}{2}\)
(b)
\(2^{x+2}-2^x=24\)Answer.
\(x=3\)
(c)
\(2^{x+2}+2^x=320\)Answer.
\(x=6\)
(d)
\(2^{x+2}-2^x=96\)Answer.
\(x=5\)
(e)
\(10^{x+1}-10^x=9000\)Answer.
\(x=3\)
(f)
\(3^{x+2}+3^x=30\)Answer.
\(x=1\)
(g)
\(4^{x+3}-4^x=63\)Answer.
\(x=0\)
(h)
\(3^{x+2}+3^x=270\)Answer.
\(x=3\)
(i)
\(3^{4x-2} - 3^{2x+1} = 486\)Answer.
\(x = 2\)
Exercise Group 13.3.7. Complex Exponents.
Solve each equation.
(a)
\(4^{\sqrt{x+1}}=2^{3x-2}\)Answer.
\(x=\frac{16}{9}\)
(b)
\(4^{-|x+1|}=\frac{1}{16}\)Answer.
\(x=-3, 1\)
