You’ve probably learned about logs in school but forgot about them. They are essentially mathematical formulas used to solve exponents.
Sometimes logs are referred to as negative exponents. Logarithmic functions are expressions useful for solving natural problems involving quantities growing exponentially.
They are really useful so read on to learn what are the reasons why logarithms are studied.
What are logarithms?
A logarithmic function is inverse to an exponential function. Expression in exponent form involves a function where a number (called the base) is raised to a certain power (called the exponent).
For instance, a base 10 raised to the power 4 equals 10,000 (10^4 = 10,000). The relationship between these three terms can be expressed in logarithmic form.
The logarithm is the exponent to which the base number must be raised to equate that number (the answer).
In our previous example, 10^4 = 10,000 can be expressed as log (base 10) of 10,000 = 4. Base 10 is a common logarithm, so it’s usually denoted as log (10,000) = 4.
For the following exponent, 4^2 = 16, the function can be denoted logarithmically as log (base 4) of 16 = 2.
To put it simply, log_b(y) = x is another way of specifying the relationship: b^x = y
What functions do logarithms perform?
Due to their specific relationship with exponential, logs are used to explore the properties of exponent functions.
A complex equation can be transformed into a simpler to solve log function. Essentially, they allow us to translate a multiplicative sequence into an additive one, thus linearizing the expression.
Logarithms are used to express large numbers conveniently. Over a large scale of values, the exponent would grow at a tremendous rate; thus, logs help express these large values more conveniently and easily.
Logs also make multiplication and division calculations simpler. When two or more numbers are multiplied, you add the logs. In contrast, when dividing numbers, you can subtract their logs.
What are the practical applications of logarithms?
Contrary to popular belief, logarithms are all around us. Most data being calculated in science and business relies on logs for flexibility and ease of display.
When data is compounding or decaying at a constant rate, logs are utilized to solve for time and other factors.
Small exponents can result in large numbers, and logarithmic scales are utilized in measuring quantities that span a large set of values.
Logs serve multiple practical applications:
The two common types of logarithms are the common and natural log. Common logs have a base of 10, while natural logs use the base of e (Euler’s number = 2.71828).
Natural logarithms solve the time in naturally decaying or growing problems. Lots of things decay naturally, for instance, when an object cools down or moving objects slow down. In such experiments, log_e is a convenient tool to help solve for time.
Other practical applications are the Richter scale used to measure earthquake intensities. The instrument measures the logarithmic intensity of quakes.
Sound is also measured on a log scale. The decibel (dB) is the logarithmic of the sound intensity.
Logs are encountered in biological sciences. For example, logs are utilized when investigating large colonies of bacteria since it will be challenging to use linear scales.
Also, logarithms make the calculations more straightforward when calculating the time for populations to grow or decay.
The pH scale is logarithmic. A pH measures the concentration of hydrogen ions within a solution. However, the concentration between pH levels varies massively, making a log scale the most convenient.
Further, measuring the free energy associated with biochemical reactions would be hard when using linear scales. As such, logarithmic functions are used.
A saving account with compound interest is another practical application of logarithms. Compounding is exponential growth.
For instance, if your savings account pays 6% annually, you can easily calculate when you will double your savings. For this example, log_1.06 (2) = 11.90. So, you’ll double your money in 12 years.
Conversely, you can calculate inflation effects on commodities. Let’s assume there’s a 4% inflation yearly, and you want to know how long it will take for goods to lose half their value. Using log_0.96 (0.5) = 16.98, it would take 17 years.
Logarithms are exponential inverses. You probably thought you’d never use logs beyond school, but it can be helpful to your savings.
Logarithms find many practical applications in physics, chemistry, and biology. Regardless of the situation, you may find logarithms being used.