Laws of logarithms and solving equations
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The teaching part (1 min) is free to watch. A free account opens the whole lecture: 4 practice questions, 2 real past-paper questions with the answers explained, the mistakes that lose marks, the full-marks tip, a one-card summary and a quick quiz.
What this lecture covers
- The slide rule trick
- What a logarithm is
- The three laws
- Change of base
Read the transcript of the teaching part
Last time
Before we start: last time, in Exponential and logarithmic graphs and models, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.
The slide rule trick
Before calculators, engineers multiplied huge numbers by adding. They used logarithms: a log turns multiplication into addition. Those laws are still the key to every exam question on this topic.
What a logarithm is
The logarithm to base a of x is the power you raise a to, to get x. So log base three of nine is two, because three squared is nine. Two special values: log base a of a is one, and log base a of one is zero. The argument must be positive. A log of zero or a negative number does not exist.
The three laws
Add logs to multiply: log of x plus log of y equals log of x y. Subtract logs to divide: log of x minus log of y equals log of x over y. A power comes down: n log x is log of x to the n. These do not work on log of x plus y. There is no law for a sum inside the log.
Change of base
To change the base of a logarithm, divide by the log of the old base, in the new base. Log base two of eight is log eight over log two. Both give three on a calculator. A neat consequence: log base a of b times log base b of a is one. They are reciprocals.
How often it comes up
How often Cambridge has asked each part, counted by ExamYard from every past paper in its question bank: Solving exponential equations · Solving logarithmic equations.
The whole chapter: Logarithmic and Exponential Functions, Simultaneous Equations.
Chapter 6: Logarithmic and exponential functions
- Exponential and logarithmic graphs and models 5 min
- Laws of logarithms and solving equations 5 min
Also for O Level Additional Maths 4037
Which topics come up most in O Level Additional Maths 4037 past papers, counted from every paper since 2004.