Change of Base Rule Explained with Simple Animations

Welcome to understanding logarithms! Today we'll explore what logarithms are and why we need different ways to calculate them.A logarithm is a special operation that finds what exponent we need to reach a certain number.Let's look at a simple example. Two raised to the third power equals eight, therefore, log base two of eight equals three.We can visualize this by seeing how the powers of two grow. Each step doubles the previous value.Here are more examples of logarithms with different bases.However, when we look at a typical calculator, we usually only find buttons for natural log, written as ln, and log base ten.This creates a challenge: How can we calculate logarithms with other bases when our calculator only has these two options?In our next section, we'll discover a powerful formula that helps us solve this problem.The change of base formula allows us to convert any logarithm to one using natural logs.Let's derive this formula step by step.We start by letting y equal log base b of x.By the definition of logarithms, this means b raised to the power y equals x.Now we take the natural logarithm of both sides.Using the logarithm property that log of a power equals the power times the log, we get y times ln of b equals ln of x.Solving for y, we divide both sides by ln of b.Therefore, log base b of x equals ln of x divided by ln of b.Let's apply this formula to convert log base 2 of 8.Using a calculator, we can compute that ln of 8 is approximately 2.0794, and ln of 2 is approximately 0.6931.Dividing these numbers gives us exactly 3.We can visualize this on a graph. The curve shows y equals 2 to the x power.The point (3,8) on this curve confirms that 2 to the third power equals 8, verifying our calculation of log base 2 of 8 equals 3.Now let's solve some practical logarithm problems using a calculator.Let's start with log base 3 of 27. Using the change of base formula, we'll convert this to natural logarithms.First, we calculate the natural log of 27 using the ln button.Next, we calculate the natural log of 3.Finally, we divide these numbers to get our answer of 3.Let's try another example: log base 5 of 125.We follow the same process, first calculating ln of 125.Then ln of 5.Dividing these gives us 3, which we can verify since 5 cubed equals 125.Let's review some common mistakes to avoid when using the calculator.Here are some tips for verifying your answers.Let's review the key points about using calculators for logarithms.Thanks for learning about practical logarithm calculations with Spark.E!