Solving Exponential and Logarithmic Equations: Algebra 2 Math Worksheet – PDF Download

Algebra 2 Exponential Log Equations Worksheet: problems 1–5

Problem 1: Same base

Solve each equation.

1. \(2^{x}=32\)
2. \(3^{x}=81\)
3. \(5^{x}=\frac{1}{125}\)
4. \(4^{x}=8\)

Problem 2: Rewrite to a common base

Solve each equation by writing both sides with the same base.

1. \(2^{3x-1}=64\)
2. \(9^{x}=27^{x-1}\)
3. \((\frac{1}{4})^{x+1}=16\)
4. \(8^{x}=\frac{1}{32}\)

Problem 3: Logarithmic to exponential form

Solve each equation by rewriting it in exponential form.

1. \(\log_{6} x=2\)
2. \(\log_{2} x=-4\)
3. \(\log_{4} x=\frac{3}{2}\)
4. \(\ln x=3\) (give a decimal to three places)

Problem 4: One-step exponential equations

Solve each equation. Round to four decimal places.

1. \(3^{x}=20\)
2. \(2^{x}=100\)
3. \(10^{x}=45\)
4. \(e^{x}=12\)

Problem 5: Isolate the power first

Solve each equation. Give exact answers when possible and decimals to three places otherwise.

1. \(4\cdot 5^{x}=100\)
2. \(3\cdot 2^{x}+5=53\)
3. \(7(1.5)^{x}=50\) (the graph of \(y=7(1.5)^{x}\) and the line \(y=50\) is shown)

Algebra 2 Exponential Log Equations Worksheet: problems 6–10

Problem 6: Exponents with a coefficient

Solve each equation. Round to three decimal places.

1. \(2^{3x}=50\)
2. \(5^{2x-1}=40\)

Problem 7: Equations with base e

Solve each equation. Round to four decimal places.

1. \(3e^{2x}=45\)
2. \(5e^{0.3x}-2=18\)

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.

1. \(A=Pe^{rt}\)
2. \(A=P(1+r)^{t}\)

Problem 9: Simple logarithmic equations

Solve each equation.

1. \(\log_{2}(x+5)=4\)
2. \(\log_{3}(2x-1)=2\)
3. \(\log(x-4)=1\)
4. \(\ln(3x)=2\) (decimal to three places)

Problem 10: Isolate the logarithm

Solve each equation.

1. \(3\log_{2} x-4=8\)
2. \(2\ln(x+1)=6\) (decimal to three places)
3. \(4\log_{5}(x-2)+3=11\)

Algebra 2 Exponential Log Equations Worksheet: problems 11–15

Problem 11: Equal logarithms

Solve each equation. Check every answer in the original equation.

1. \(\log_{5}(3x+1)=\log_{5}(x+9)\)
2. \(\ln(2x-3)=\ln(x+5)\)
3. \(\log_{2}(x^{2})=\log_{2}(5x-6)\)

Problem 12: Add two logarithms

Solve each equation. The graph of \(y=\log_{2} x+\log_{2}(x-6)\) is shown for part (a).

1. \(\log_{2} x+\log_{2}(x-6)=4\)
2. \(\log_{3} x+\log_{3}(x+6)=3\)

Problem 13: Subtract two logarithms

Solve each equation.

1. \(\log_{3}(x+8)-\log_{3} x=2\)
2. \(\ln(x+5)-\ln(x-1)=\ln 3\)

Problem 14: Reject extraneous solutions

Solve each equation and name any extraneous solution.

1. \(\log x+\log(x-3)=1\)
2. \(\log_{4}(x+2)+\log_{4}(x-1)=1\)

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.

Algebra 2 Exponential Log Equations Worksheet: problems 16–20

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.

Algebra 2 Exponential Log Equations Worksheet: problems 21–23

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.

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