These problems practice solving exponential equations with a common base or with logarithms, solving logarithmic equations, and checking for extraneous solutions. The last problems apply the methods to interest, population, half-life, earthquakes and pH.
Problems run from easy to hard and match Common Core standards F-LE.A.4 and A-SSE.B.3c at the Algebra 2 level. Round only at the final step.
Problem 1: Same base
Solve each equation.
- \(2^{x}=32\)
- \(3^{x}=81\)
- \(5^{x}=\frac{1}{125}\)
- \(4^{x}=8\)
Problem 2: Rewrite to a common base
Solve each equation by writing both sides with the same base.
- \(2^{3x-1}=64\)
- \(9^{x}=27^{x-1}\)
- \((\frac{1}{4})^{x+1}=16\)
- \(8^{x}=\frac{1}{32}\)
Problem 3: Logarithmic to exponential form
Solve each equation by rewriting it in exponential form.
- \(\log_{6} x=2\)
- \(\log_{2} x=-4\)
- \(\log_{4} x=\frac{3}{2}\)
- \(\ln x=3\) (give a decimal to three places)
Problem 4: One-step exponential equations
Solve each equation. Round to four decimal places.
- \(3^{x}=20\)
- \(2^{x}=100\)
- \(10^{x}=45\)
- \(e^{x}=12\)
Problem 5: Isolate the power first
Solve each equation. Give exact answers when possible and decimals to three places otherwise.
- \(4\cdot 5^{x}=100\)
- \(3\cdot 2^{x}+5=53\)
- \(7(1.5)^{x}=50\) (the graph of \(y=7(1.5)^{x}\) and the line \(y=50\) is shown)
Problem 6: Exponents with a coefficient
Solve each equation. Round to three decimal places.
- \(2^{3x}=50\)
- \(5^{2x-1}=40\)
Problem 7: Equations with base e
Solve each equation. Round to four decimal places.
- \(3e^{2x}=45\)
- \(5e^{0.3x}-2=18\)
Problem 8: Solve a formula for time
Solve each formula for \(t\). Then find \(t\) when \(P=2000\), \(A=3500\) and \(r=0.05\), to two decimal places.
- \(A=Pe^{rt}\)
- \(A=P(1+r)^{t}\)
Problem 9: Simple logarithmic equations
Solve each equation.
- \(\log_{2}(x+5)=4\)
- \(\log_{3}(2x-1)=2\)
- \(\log(x-4)=1\)
- \(\ln(3x)=2\) (decimal to three places)
Problem 10: Isolate the logarithm
Solve each equation.
- \(3\log_{2} x-4=8\)
- \(2\ln(x+1)=6\) (decimal to three places)
- \(4\log_{5}(x-2)+3=11\)
Problem 11: Equal logarithms
Solve each equation. Check every answer in the original equation.
- \(\log_{5}(3x+1)=\log_{5}(x+9)\)
- \(\ln(2x-3)=\ln(x+5)\)
- \(\log_{2}(x^{2})=\log_{2}(5x-6)\)
Problem 12: Add two logarithms
Solve each equation. The graph of \(y=\log_{2} x+\log_{2}(x-6)\) is shown for part (a).
- \(\log_{2} x+\log_{2}(x-6)=4\)
- \(\log_{3} x+\log_{3}(x+6)=3\)
Problem 13: Subtract two logarithms
Solve each equation.
- \(\log_{3}(x+8)-\log_{3} x=2\)
- \(\ln(x+5)-\ln(x-1)=\ln 3\)
Problem 14: Reject extraneous solutions
Solve each equation and name any extraneous solution.
- \(\log x+\log(x-3)=1\)
- \(\log_{4}(x+2)+\log_{4}(x-1)=1\)
Problem 15: Quadratic form
Solve \(2^{2x}-6\cdot 2^{x}+8=0\). Use the substitution \(u=2^{x}\). The graph of \(y=4^{x}-6\cdot 2^{x}+8\) is shown.
Problem 16: Quadratic form with e
Solve \(e^{2x}-5e^{x}+6=0\). Round to four decimal places.
Problem 17: Equivalent growth rates
A town has population \(P(t)=12000(1.035)^{t}\), where \(t\) is in years. Write \(P\) in the form \(12000e^{kt}\), and find the equivalent growth rate per month as a percent to three decimal places.
Problem 18: Bacteria growth
A colony starts with 200 cells and grows 8% each hour, so \(N(t)=200(1.08)^{t}\). When does it reach 1,000 cells? Find the doubling time too. The graph with the line \(N=1000\) is shown.
Problem 19: Compound interest
Aisha invests \$3,000 at 4.5% compounded quarterly, so \(A=3000(1+\frac{0.045}{4})^{4t}\). How many years until the account reaches \$5,000?
Problem 20: Continuous compounding
Liam deposits \$1,500 at 3.2% compounded continuously, so \(A=1500e^{0.032t}\). Find the time to double, and the time to reach \$4,000.
Problem 21: Medicine half-life
A patient takes a 200 mg dose of a drug with half-life 6 hours, so \(N(t)=200(\frac{1}{2})^{t/6}\). When do 15 mg remain? The graph with the line \(N=15\) is shown.
Problem 22: Earthquake comparison
Magnitude is \(M=\log\frac{I}{I_{0}}\). Noah compares two earthquakes. (a) How many times as intense is a magnitude 7.6 quake as a magnitude 5.2 quake? (b) A quake is 250 times as intense as a magnitude 5.2 quake. Find its magnitude to the nearest tenth.
Problem 23: pH and decibels
(a) Sofia measures a drink with pH 3.8. Find \([H^{+}]\) from \(\text{pH}=-\log[H^{+}]\), in scientific notation with three significant digits. (b) A siren is 95 dB. Find \(\frac{I}{I_{0}}\) from \(L=10\log\frac{I}{I_{0}}\), in scientific notation with three significant digits.
See every chapter on the Algebra 2 math page.