#Log base 2 of 4 how to

To do it this way, to say that 'x' is the power you raise 3 to to get to 81, you had to use algebra here, while with just a straight up logarithmic expression, you didn't really have to use any algebra, we didn't have to say that it was equal to 'x', we could just say that this evaluates to the power I need to raise 3 to to get to 81. But you didn't necessarily have to use algebra. Why is a logarithm useful? And you'll see that it has very interesting properties later on. So if you want to, you can set this to be equal to an 'x', and you can restate this equation as, 3 to the 'x' power, is equal to 81. What would this evaluate to? Well this is a reminder, this evaluates to the power we have to raise 3 to, to get to 81. So with that out of the way let's try more examples of evaluating logarithmic expressions. This is saying 'hey well if I take 2 to some 'x' power I get 16'.' This is saying, 'what power do I need to raise 2 to, to get 16 and I'm going to set that to be equal to 'x'.' And you'll say, 'well you have to raise it to the fourth power and once again 'x' is equal to 4. Log, base 2, of 16 is equal to what, or is equal in this case since we have the 'x' there, is equal to 'x'? This and this are completely equivalent statements. Now the way that we would denote this with logarithm notation is we would say, log, base- actually let me make it a little bit more colourful. And this is what logarithms are fundamentally about, figuring out what power you have to raise to, to get another number. So for example, let's say that I start with 2, and I say I'm raising it to some power, what does that power have to be to get 16? Well we just figured that out. We know that we get to 16 when we raise 2 to some power but we want to know what that power is. But what if we think about things in another way. 2 multiplied or repeatedly multiplied 4 times, and so this is going to be 2 times 2 is 4 times 2 is 8, times 2 is 16.

If I were to say 2 to the fourth power, what does that mean? Well that means 2 times 2 times 2 times 2.

#Log base 2 of 4 how to

So we already know how to take exponents. Let's learn a little bit about the wonderful world of logarithms.

They are an essential mathematical concept with diverse applications in the real world. Overall, logarithms provide a powerful toolset for dealing with exponential relationships, making computations more manageable, and offering insights into various phenomena across different disciplines. Machine Learning and Data Science: Logarithms find applications in various machine learning algorithms and data analysis tasks, like feature scaling, transforming skewed distributions, and handling large ranges of numerical data. Probability and Statistics: Logarithms are used in probability and statistics to transform skewed data or convert multiplicative relationships into additive ones, making them more amenable to certain statistical techniques. This simplifies the analysis of signals in various applications. They enable lossless compression techniques that can later be reconstructed to the original data.įrequency Analysis: Logarithms are used in signal processing and frequency analysis to convert exponential growth or decay relationships into linear ones. They help determine how an investment grows over time, accounting for interest being added to the initial investment and the interest already earned.ĭata Compression: Logarithms are employed in data compression algorithms to reduce the size of data for efficient storage and transmission. It is based on logarithms and provides a more intuitive representation of relative loudness or signal strength.Ĭompounding Interest: Logarithms are used in finance to calculate compound interest. For example, the Richter scale uses logarithms to quantify the energy released by earthquakes, and the pH scale uses logarithms to measure the acidity of a solution.ĭecibel Scale: In acoustics and electronics, the decibel scale is used to express the ratio of two quantities, such as sound or power levels. Measuring Relative Magnitudes: Logarithms allow us to express large ranges of numbers in a more manageable form. Multiplication and division of numbers can be converted to addition and subtraction, respectively, using logarithmic properties. Simplifying Complex Calculations: Logarithms can simplify computations, especially when dealing with large numbers or complicated mathematical operations. This is particularly useful when dealing with exponential growth or decay problems. When you have an equation of the form a^x = b, taking the logarithm of both sides allows you to solve for x. Solving Exponential Equations: Logarithms provide a way to solve equations involving exponents. Logarithms serve several important purposes in mathematics, science, engineering, and various fields.