# C2: Change of Base of Logarithms Learning Objective: to understand that the base of a logarithm may need changing to be able to solve an equation.

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C2: Change of Base of Logarithms Learning Objective: to understand that the base of a logarithm may need changing to be able to solve an equation

Changing the base of a logarithm Suppose we wish to calculate the value of log 5 8. We can’t calculate this directly using a calculator because it only find logs to the base 10 or the base e. We can change the base of the logarithm as follows: Let x = log 5 8 So:5 x = 8 Taking the log to the base 10 of both sides: log 5 x = log 8 x log 5 = log 8 So: 1.29 (to 3 s.f.)

Changing the base of a logarithm In general, to find log a b : Let x = log a b, so we can write a x = b Taking the log to the base c of both sides gives: log c a x = log c b x log c a = log c b So:

Example : We can use the change of base of logarithms to solve equations. For example: Find, to 3 significant figures log 8 11. We can solve this by changing to base 10: log 8 11 = log 10 11 / log 10 8 Using a calculator: x = 1.15 (to 3 s.f.)

Task 1 : Find to 3 d.p. log 7 120 log 3 45 log 2 19 log 11 3 log 6 4 Solve 8 x = 14, 9 x = 99, 12 x = 6

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