This means we have two forms for the same thing: Log Form: Log ax is the exponent to which the base a must be raised to obtain x. We will set our exponent of y equal to the base a logarithm of x: How can we do this? Up to this point, we have not seen any method that allows us to solve an equation for the exponent. ![]() Let's begin with the exponential function.Īt this point, we want to solve the equation for y. We can derive this function by following the steps needed to find the inverse of a function. This is because it doesn"t matter, as long as they are both the same.In the last lesson, we learned about the exponential function.Ī logarithmic function is the inverse of the exponential function. Notice how we haven"t said what the base is. Logarithms can be used to help solve equations of the form a x = b by "taking logs of both sides". If we are given equations involving exponentials or the natural logarithm, remember that you can take the exponential of both sides of the equation to get rid of the logarithm or take the natural logarithm of both sides to get rid of the exponential. It therefore follows that the integral of 1/x is ln x + c. Ln x is also known as the natural logarithm. Most calculators can only work out ln x and log 10x (usually just written as "log" on the button) so this formula can be very useful. This is a very useful way of changing the base (in this formula, the base does matter!). This is because for the laws of logarithms, it doesn"t matter what the base is, as long as all of the logs are to the same base.Īnother important law of logs is as follows. NB: In the above example, I have not written what base each of the logarithms is to. The properties of indices can be used to show that the following rules for logarithms hold: Remember that e is the exponential function, equal to 2.71828… It is generally recognised that this is shorthand: You may often see ln x and log x written, with no base indicated. Logarithms are another way of writing indices. Notice that lnx and e x are reflections of one another in the line y = x. In the diagram, e x is the red line, lnx the green line and y = x is the yellow line. Like most functions you are likely to come across, the exponential has an inverse function, which is log ex, often written ln x (pronounced 'log x'). ![]() In other words:īecause of this special property, the exponential function is very important in mathematics and crops up frequently. The exponential function, written exp(x) or e x, is the function whose derivative is equal to its equation.
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