Unit 4 Mathematics Aims Introduce logarithms applications Objectives

Unit 4: Mathematics Aims • Introduce logarithms applications. Objectives • Use laws of logarithms to solve given equations. 1

Logarithms • Logarithms to various bases: red is to base e, green is to base 10, and purple is to base 1. 7. • Each tick on the axes is one unit. • Logarithms of all bases pass through the point (1, 0), because any number raised to the power 0 is 1, and through the points (b, 1) for base b, because a number raised to the power 1 is itself. The curves approach the y-axis but do not reach it because of the singularity at x = 0.

• In other words, log(10) is 1 and log(100) is 2 (because 10² = 100). • Logs can easily be found for either base on your calculator. Usually there are two different buttons, one saying "log", which is base ten, and one saying "ln", which is a natural log, base e. It is always assumed, unless otherwise stated, that "log" means log 10.

Review Log rules from last week • Logc (am) = m logc(a) • Example log 2 X = 8 28 = X X = 256 • 10 log x = X • “ 10 to the” is also the anti-log (opposite)

Example 2 Review Log rules • Example 2 log X = 0. 25 • Raise both side to the power of 10 10 log x = 100. 25 X = 1. 78 10 log Is canceled out by raising the other side by 10

Example 3 Review Log Rules • Solve for x 3 x = 1000 • Log both sides to get rid of the exponent log 3 x = log 1000 x log 3 = log 1000 x = log 1000 / log 3 x = 6. 29

• Solve the equation = 15. • We can solve this by taking logarithms of both sides. • Log = log 15 • Now using the laws of logarithms, and in particular log = n log A, the left hand side can be • re-written to give • x log 4 = log 15 • This is more straightforward. The unknown is no longer in the power. Straightaway, dividing both sides by log 4, • x = log 15 log 4

Solving equations using Logarithms • When we introduce numbers into this equation - 5 = 15 we can log both sides. • Log 5 = log 15 from here we can apply algebra rules and divide both sides to leave x = log 5 = 0. 594 • log 15