Logarithmic and Exponential Functions: worked exam questions
13 solved logarithmic and exponential functions questions, each worked line by line the way the 0606 mark scheme rewards. Learn the method in the Logarithmic and Exponential Functions topic notes, then see it applied here.
A Hidden Quadratic in an Exponential Equation
Solve 3^(2x) − 10(3^x) + 9 = 0 with the substitution y = 3^x, a 0606 question that turns a scary exponential into an ordinary quadratic.
A Hidden Quadratic in Logarithms
Solve (log₃x)² − log₃x² = 3 by substituting u = log₃x to reveal a quadratic, then converting back, a common 0606 logarithm question.
An Exponential Growth and Decay Problem
Apply an exponential model to find a value and solve for time using ln, a 0606 question that puts e^x and logarithms in a real context.
Exponential Decay: Finding a Time
Use an exponential decay model to find when a quantity halves, a 0606 question applying ln to a negative exponent in a real context.
Expressing a Logarithm in Terms of Given Logs
Given log_a 2 = p and log_a 3 = q, express log_a 4.5 in terms of p and q, a 0606 question testing fluent use of the three log laws in reverse.
Reducing y = Ab^x to Linear Form
Given a straight-line graph of lg y against x, find A and b in y = Ab^x, a 0606 question on taking logs to linearise a relationship and read off the constants.
Simplifying an Expression with Log Laws
Combine lg 8 + lg 5 − lg 4 into a single logarithm and evaluate it, a 0606 question testing fluent use of the addition and subtraction laws.
Solving a Log Equation with Terms on Both Sides
Solve lg(x + 2) = 1 + lg(x − 1) by collecting logs and handling the lone constant, a 0606 question testing the subtraction law and base-10 conversion.
Solving an Equation with Different Bases
Solve 2ˣ = 3ˣ⁻¹ by taking logs of both sides and collecting the x terms, a 0606 exponential equation where the bases don't match.
Solving an Equation with the Laws of Logarithms
Solve log₃(x + 1) + log₃(x − 1) = 1 by combining logs, then reject the invalid root, a 0606 question where checking the domain earns the final mark.
Solving an Exponential Equation with ln
Solve 5e^(2x) = 40 exactly and as a decimal, a 0606 question on taking natural logs to bring the power down, with the exact-form mark explained.
Solving Simultaneous Logarithmic Equations
Solve a pair of simultaneous equations in log₂x and log₂y by treating the logs as the unknowns, then converting back, a neat 0606 logs question.
Using Logs to Solve an Exponential Equation
Solve 2^x = 7 by taking logs and using the power law, a 0606 question on bringing the exponent down when the base isn't e.
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