Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
The last six problems use savings, medicine, earthquakes, sound and a flu outbreak.
Standards: F-LE.A.4, F-BF.B.5, F-IF.C.8b · Level: Grade 12 Precalculus
Problem 1 Switch Between Forms ★★★
Rewrite each statement in the other form. (a) 63 = 216 (b) log9 81 = 2 (c) 10−2 = 0.01
Problem 2 Evaluate Logarithms Exactly ★★★
Evaluate each without a calculator. (a) log2 64 (b) log5(1/25) (c) log9 3 (d) ln e4
Problem 3 Expand a Logarithm ★★★
Expand log3 27a2b using the laws of logarithms.
Problem 4 Condense a Logarithmic Expression ★★★
Write 2 ln 5 − ln 10 + ln 4 as a single logarithm and simplify.
Problem 5 Match the Bases ★★★
Solve each equation. (a) 2x = 32 (b) 3x + 1 = 81
Problem 6 Read a Logarithmic Graph ★★★
The graph shows y = log2 x. Use it to find log2 4, solve log2 x = 3, and state the domain and the vertical asymptote.
Problem 7 Change of Base ★★★
Use the change-of-base formula to approximate log7 50 and log4 0.3 to three decimal places.
Problem 8 Exponential Equation with Logs ★★★
Solve 7x − 2 = 19. Give the exact answer and a decimal to three places.
Problem 9 Isolate a Natural Exponential ★★★
Solve 3e0.4x = 21.
Problem 10 Logarithmic Equation with a Check ★★★
Solve log3(x + 1) + log3(x − 5) = 3 and check each candidate.
Problem 11 Natural Log Equation ★★★
Solve ln(2x − 1) = 3. Give an exact answer and a decimal to three places.
Problem 12 Inverse Pairs and Reflections ★★★
The graph shows f(x) = 3x and its inverse g(x) = log3 x. Given that f(2) = 9, name a point on g, then find g(27) and the range of g.
Problem 13 Solve with a Fractional Exponent ★★★
Solve 6 · 2x/3 = 96.
Problem 14 Doubling Culture ★★★
A culture starts with 500 cells and doubles every 3 hours, so N(t) = 500 · 2t/3. Find when the culture reaches 8,000 cells and when it reaches 20,000 cells.
Problem 15 Logistic Starting Value and Limit ★★★
A population follows L(t) = 5001 + 9e−0.5t (t in years). Find the starting population, the carrying capacity, and the time when the population reaches 250.
Problem 16 Continuously Compounded Savings ★★★
A savings account holds $2,500 and earns 4.2% interest compounded continuously, so A = 2500e0.042t. Find the balance after 8 years and the time needed to reach $4,000.
Problem 17 Medicine Half-Life ★★★
A 200 mg dose has a half-life of 6 hours, so A(t) = 200(1/2)t/6. When are 25 mg left? When are 60 mg left?
Problem 18 Comparing Earthquakes ★★★
Magnitude is M = log(A/A0), where A is the wave amplitude. How many times larger is the amplitude of a magnitude 6.8 quake than that of a magnitude 5.3 quake? What magnitude has an amplitude of 5,000A0?
Problem 19 Sound Levels ★★★
Sound level in decibels is L = 10 log(I/I0). A concert measures 105 dB and a conversation 60 dB. How many times greater is the concert’s intensity? By how many decibels does the level rise when two identical sources play together?
Problem 20 Flu Spread in a School ★★★
In a school of 800 students, the number infected after t days is N(t) = 8001 + 49e−0.35t. When are 400 students infected, and when do 720 (90%) become infected?
