![]() ![]() ![]() In calculus, we will make extensive use of the natural exponential function y=e x, because its derivative is the same function. Solve the following inequalities, giving your answers correct to 3 significant figures: d However, if the base is less than 1, the inequality sign will be reversed (because the log function in this case is a decreasing function).Īlso, remember that dividing or multiplying by a negative reverses the inequality sign (any argument less than 1 gives a negative because log1=0). Solve the following equations, giving your answers to 3 decimal places:Īs with solving exponential equations we take the logarithm of both sides of the equation (to ensure the variable is not in the index). Substituting and then solving related quadratic equations.Taking the logarithm of each side of the equation and then using algebraic manipulation to isolate the variable and.The two main “new” methods we will use are: Using logarithms to solve exponential equations with different bases We should test solutions to the original equations to ensure they satisfy the conditions (i.e. If possible we should try to get all terms in the same base, so for instance if we have log 27+5=log 2x, we can change the 5 into log 232 to make things easier. We’ve already been doing this, but now we can start to make them more complicated by relying on our logarithm laws. Write in terms of log ax, log ay and log az:.As well as understanding the derivation, you should immediately recognise when one of these laws can be applied. These are very important laws that will be used extensively across P3 topics and so should be memorised immediately. Rewrite log 3x=4 in exponential form and solve īy using the earlier identity and replacing x with a log ax (as per previous identity) we can derive the following laws: Worked Examples using any base logarithms Some obvious identities that must nevertheless be memorised are as follows: This is defined provided that a>0, a is not equal to 1 and x>0 (why are these conditions necessary?) Obviously the above logic applies not for just base 10, but for other bases. We can find a numerical solution using the calculator button: ] We can read this as saying that x is the number that 10 must be raised to to get 15. The logarithmic function lets us calculate 10 x=15 directly, as: 10 x=15 x=log 1015. An exponential function has the form f(x)=a x ![]()
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