Solving logarithmic equations using definition
The simplest way of solving logarithmic equations is by comparing two logs. This method takes into account two logs with the same base. Take for example, log5 (b) =log5 (18), here the base of both the logs is 5 and thus by comparing one can see that b=18. This is one of the easiest methods involving logarithmic equations. Let us take another example involving a quadratic equation: log2(y2)=log2(y2-1) since we have the same base 5 on either side we simply equate the two quadratic equations and solve for y. On simplifying, we get the solution as y=1. Also, note that logarithms cannot be taken for a negative value and thus there always has to be a non-negative value while working with logarithms.
Solving logarithmic equations using exponentials
When it comes to solving logs using exponentials, you will come across a lot of powerpoint presentation s on the internet as well as web forums which will help you develop a better understanding of the topic.
A lot of engineering paper s have been written and made available on the internet on this topic that will help you develop an in-depth knowledge of the concept and provide you quick and easy tips to deal with logarithms.
Previously we dealt with equations having log on either side, but the rule changes a little bit when the second side of the equation contains a numerical value instead of a logarithmic equation. Let us see how. Take for example, the equation log3 (a) =2. In order to obtain the value of a, the base of the log that is 3, shifts to the other side with its power raised to the number on that side. Thus, the answer would be a=3^2 or a=9. Similarly, let’s take the following case where log3 (27) =a. Now the same rule applies here too and we shift the base which is 3 on the other side. So, 27=3^a or 27=3^3 and thus, by comparison, we can see that a=3.
writing assignments on logarithmic equations and solving as many questions as possible. By doing so, you will be able to have a strong grip on the subject and can solve long and complex maths problems involving log with great ease. Take the help of the internet and practice exercises on log. Learn how to look at a log and an antilog table and get your answers without making use of a calculator for various logarithmic values.
Basic log rules for solving logarithmic equations
Apart from simplifying the various logarithmic equations, there are also some direct formulae which when applied can help you fetch direct solutions in many cases. The three most basic rules associated with logarithm that needs to be memorized forever are:
- Logx(ab)=logx(a)+logx(b); the product rule
- Logx(a/b)= logx(a)-logx(b); the quotient rule
- Logx(a^b)=b logx(a); the exponent rule
The first rule deals with multiplication using log while the second one takes into account division for logarithms and the last one being solely for exponents. Also, note that the bases of both the logs must be the same while applying these formulae and they do not hold true for cases like logx(a)+logy(b) where the bases ‘x’ and ’y’ are of different values each. These logarithmic rules are like the three pillars of logarithm. Without them, it is almost impossible for anyone to solve various complex logarithm problems.
Just as the first rule for writing a descriptive essay is to make sure that the information that you are intending to use is valid and applies to the topic that you are writing on. The concept should be easy to explain, but you will need to look at the format in which you are writing the essay since it needs to be proper. As college students, we are expected to know how to write a speech which does not lose out on its impact factor. If your essay requires you to write about formulae, then you better have a justifiable and detailed explanation as to why the formula came into being and what are the elements that have been considered in computing the formula. Once each of the elements has been explained appropriately, you will need to explain to the readers how the different elements in the essay pertain to the topic of discussion and how the value that you will be getting from that formula will be precise. In this way, you will be able to explain how solving logarithmic equations will become easier no matter how complex it seems in the beginning.
Solving logarithmic equations by expanding
you can use logarithms to solve for values and even expand expressions. This means that when the log expression gets a little complicated, all you can do is split this expression using those three formulae and simplify them further. For example, take the following expression: log2 (8/a), here the base of the expression is 2 and we apply the formulae and split it as:
Here are a few very important derived expressions about logarithms that you need to keep in mind before you begin solving logarithmic equations.
- Also note that loga (a) is always equal to 1, that is the logarithm of a similar base always fetches 1 as an answer.
- Another important thing worth mentioning is that: loga (negative number) or loga (zero) = undefined and does not hold true for any equation whatsoever.
- Make a note that whenever you see a log without a base written to it, in that case, we take 10 to be the base for our log.
so for example;
[As log 10(10) =1]
- Similarly log1000=3, log10000=4 and so on.
- The value of log1 is always zero.
Solving logarithmic equations using a calculator
The most important concept while dealing with logs is that of ‘natural logarithm’ or ‘ln’. It has a base ‘e’ which has a value of nearly 2.7 and thus requires the use of a calculator when problems related to natural logarithms arise. Solve for example, the following equation ln(x)=3. Now for such an expression also the same rule of log applies, but with a base ‘e’. Thus, it becomes, x=e^3 or 2.7^3.
On calculating the result with the help of a calculator we see that the answer is somewhere around 20.087. Let us take another example where you will require a calculator to come to a conclusion.
The equation is:
- Log 2(4y)=5.1;
Why do we need logarithmic equations
Basically, logarithms are nothing but a particular solution for the exponential equations. Whenever we come across questions like, after how much time will the population of a town double or if the half-life of an element is this, then after what time will it decay; we make use of logarithms. Compound interest is also another area where complex exponential sums can be solved simply within minutes by converting both sides of the equation to logarithm and then taking antilog on both sides after the final calculations have been done.
Another reason why historians preferred logarithm was that logarithm takes into account the addition of two values which were earlier a product (see product rule) and thus dealing with the addition becomes much easier than dealing with the complete multiplication. Apart from its computational “trick”, it is the foundation of mapping technique as described by Christian Blatter and makes use of the concept of self-adjoint generators which has many mathematical applications and helps deal in quantum mechanics by relating the physical observations and symmetrical properties together.
Logarithm and its invention
Years before the calculator was invented, the invention of logarithm by John Napier proved to be a great labour-saving device. Although, many claim that logarithm was an invention of the 8th century, however, its use for calculation and various computational purposes is attributed to the great Scottish man named John Napier. It is said that along with an oxford professor named Mr. Henry, Napier constructed a logarithmic table in base 10 and its use in the decimal point. The anti-logarithmic tables were invented long after that. In fact, logarithmic tables were even called the earliest calculating machines of a time where calculator seemed like an invention of the future generation.
In every walk of our lives, logarithmic equations are involved in some way or the other and those who are doing research on such topics understand the role of logarithmic functions in a better way than others. College students who are pursuing mathematics degrees are usually given assignments on logarithmic functions and there is a lot of scope for research in the field. Based on the progress we have made, hopefully, we will go even deeper into the concept and comprehend it better.
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