Before You Watch

This video introduces solving equations where the letter you are asked to find the value of is in the index (also called the exponent). This requires the use of logarithms. The definition of a logarithm is, if:

ab= c


logac = b

When we read the line above, we say “log to the base ‘a’ of ‘c’, is equal to 'b'”.

Another way we can think about logarithms is as the inverse of the exponential: for example, if we have a number x, and we put a number, say, 5 to the power of x.


and then we put that to the log of base 5:


that is equal to x:

log5(5x) = x

In other words, if we start with x, and then put 5 to the power of x, we can think of the log process as “undoing” the process of putting 5 to the power of x. So these two processes are the inverse of each other, in a similar way to multiplication and division being the inverse of each other (that is, if you start with x, then multiply by 5, then divide by 5, you get back to x).

Need more of an introduction to the nature of logs? Look at one of the links below before watching this video. ,


As well as having an understanding of the nature of logs, it is useful to revise your knowledge of the number “e” before watching this video. Remember, “e” is an irrational number that has many real world applications, and that's why it has been given a special name. In fact, “e” is similar to the number π, which is also an irrational number that has a special name because of its applications. The number “e” is approximately equal to 2.718, however, most calculators have a dedicated button for it.

Comfortable with using logarithms and the number “e”? Then you're ready for this video!

The Video

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Some Practice Questions


Now What?

This video shows you how to solve an important category of equations, where the letter is in an index. It will also have refreshed your memory of logarithms and the number “e”.

Once you are confident in your ability to solve equations like this, check your skill at solving other types of equations, such as Quadratic Equations or Simultaneous Equations.

But When I am Going to Use This

Exponentials are often found in situations where the rate that something grows is related to the size it already is. A very simple example of this is a home loan: the bigger the loan or interest rate, the faster it grows. This type of growth is found naturally as well, for instance, in bacterial growth.

Logarithms are also commonly used to measure things over a very large scale, where the values getting measured can be very large and very small. For example, the decibel, the standard unit for measuring the volume of sounds, is a logarithmic scale. Another example of a logarithmic scale is the Richter scale, the standard scale for the measurement of the strength of earthquakes.

Other Links

Fort Bend Tutoring features a series of YouTube videos covering many different mathematical concepts. The video below covers solving exponential equations, and also demonstrates how to solve exponential equations in situations where you don’t need to use the logarithms.

The Khan Academy has a comprehensive set of video tutorials covering a large range of mathematical and other concepts, as well as questions to test your knowledge. Look at the top of this resource for the link to the chapter that covers logarithms. Half way through the chapter is a video that shows you how to solve the exponential equation.

Patrick JMT (Just Maths Tutorials) has a comprehensive set of video tutorials covering a large range of mathematical concepts. The video linked below demonstrates how to solve exponential equations.