Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
They move from converting forms to modeling with pH, decibels and earthquake magnitude. Work the first ten without a calculator wherever possible and keep every step of the algebra.
Standards: F-LE.A.4, F-BF.B.4a, F-IF.C.7e · Level: Grade 11 Algebra 2
Problem 2 Logarithmic to exponential form ★★★
Write log6 216 = 3 in exponential form.
Problem 3 Evaluate a basic logarithm ★★★
Evaluate log3 243.
Problem 4 Logarithm of a power of 10 ★★★
Evaluate log 0.001 without a calculator.
Problem 5 Find the base from a graph ★★★
The graph below shows y = logb x. Use the point (4, 1) to find b, then give the value of logb 64.
Problem 6 Logarithm with a fractional base ★★★
Evaluate log1/2 8.
Problem 7 Logarithm giving a fraction ★★★
Evaluate log25 5.
Problem 8 Natural logarithm of a power of e ★★★
Evaluate ln e4 and ln 1e.
Problem 9 Expand a logarithm ★★★
Expand log3(9xy2) so that no product, quotient or power remains inside a log.
Problem 10 Condense a logarithmic expression ★★★
Write 2 log5 m + log5 4 − log5 n as a single logarithm.
Problem 11 Change of base ★★★
Use the change-of-base formula to approximate log3 50 to the nearest hundredth.
Problem 12 Domain and asymptote ★★★
The graph below is y = log2(x − 3). State the domain, the equation of the vertical asymptote and the x-intercept.
Problem 13 Locate a log on a number line ★★★
On the number line, log 5000 lies between two consecutive integers. Name them, then use a calculator to give log 5000 to the nearest tenth.
Problem 14 Reflect points to find an inverse ★★★
The graph of y = 3x contains (0, 1), (1, 3) and (2, 9). List the corresponding points of the inverse and write the inverse function.
Problem 15 Earthquake energy comparison ★★★
Earthquake magnitude M is a base-10 log of shaking amplitude. A quake of magnitude 7.1 is compared with one of magnitude 5.9. How many times larger is the amplitude of the stronger quake? Round to the nearest hundredth.
Problem 16 Decibel level of a vacuum cleaner ★★★
Sound level in decibels is L = 10 log(II0), where I0 = 10−12 W/m². A vacuum cleaner produces an intensity I = 3.2 × 10−5 W/m². Find its sound level to the nearest tenth.
Problem 17 pH of a soil sample ★★★
The pH of a liquid is −log[H+], where [H+] is in moles per liter. A rainwater sample has [H+] = 2.5 × 10−6. Find its pH to the nearest hundredth. Is it acidic (pH below 7)?
Problem 18 Inverse of a logarithmic function ★★★
Let f(x) = 2 log5 x + 1. Find f−1(x) and verify it by computing f−1(f(25)).
Problem 19 Using given log values ★★★
Suppose logb 2 = 0.43 and logb 3 = 0.68. Find logb 18 and logb(89) using the properties of logarithms.
Problem 20 Justify a change-of-base consequence ★★★
Show that log8 32 = 53 by rewriting both numbers as powers of 2 and using the change-of-base idea.
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