Log & Exponential Intersections and One-Root Proofs

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1 session

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About

In this one-session class we use graphs of exponential, logarithmic, and polynomial functions to analyse equations and prove how many real roots they have. Working from A-level style questions, we will sketch curves such as y = e^x − 1 or y = ln(x + 1) and compare them with straight lines or other curves. By sketching both sides of each equation on the same axes and reasoning about how many times the graphs can intersect, we show that the equation has exactly one real solution. This is a great way to connect algebra, graphs, and rigorous exam-style arguments.
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A recent graduate of CAIE A-levels with Mathematics, Physics, and Computer Science. Currently, pursuing my B.Sc. in Computer Science & Engineering. Passionate about math and education, I am eager to help others excel in subjects including math, physics, computer science, and SAT Prep. Excited to tutor in various math areas such as Algebra, Geometry, Pre-Calculus, Calculus, and Statistics. Let's learn and grow together!

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✋ ATTENDANCE POLICY

Please join on time and stay for the whole session so you can follow the full reasoning from sketch to conclusion. You’ll be asked to sketch graphs on paper and share your ideas in the chat or by speaking. If you know you cannot attend or will miss most of the class, please unregister to free the seat for someone else.

SESSION 1

5

Dec

SESSION 1

Logarithms

Fri 4:15 AM - 5:15 AM UTCDec 5, 4:15 AM - 5:15 AM UTC

In this session we will practise using graphs of logarithmic, exponential, and polynomial functions to study equations. Typical tasks include:
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• Sketching y = e^x − 1 and similar curves and comparing them with a line.  
• Sketching y = ln(x + 1) and using it with a cubic like y = 40 − x^3 to argue that the equation x^3 + ln(x + 1) = 40 has exactly one real root.  
• Sketching functions such as y = 2 − x and y = ln x, or y = 5e^(−x) and y = √x, and using their relative positions to show that each equation has a single solution.  
• Using careful reasoning about monotonicity and end behaviour to support the graphical arguments.
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We’ll go step by step: identify each side of the equation, sketch both curves, mark key intercepts and behaviour, and then justify the number of intersections. Please have graph paper, pen, and a calculator ready.
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Dec 5

1 week

60 mins

/ session

Next session on December 5, 2025

SCHEDULE

Friday, Dec 5

4:15AM