Log Rules

In order to manipulate and simplify logarithmic expressions, log rules are crucial mathematical tools. The rules of logarithms are directly derived from the rules of exponents. 
Four primary logarithmic rules are commonly applied:

• Product Rule: logb (𝑚𝑛) = logb 𝑚 + logb 𝑛

• Quotient Rule: logb (mn) = logb 𝑚 - logb 𝑛 

• Power Rule:  logb (𝑚n) =  𝑛 logb 𝑚

• Change of Base: loga b = logc blogc a

These guidelines are particularly helpful for solving logarithmic equations and simplifying complex logarithmic expressions. 

Furthermore, the relationship between exponential and logarithmic forms (bx=m ⇔ logbm=x) yields some basic identities:

• Since b0=1, we get logb1=0

• Since b1=b, the result will be logbb=1

Working with logarithms in algebra and more complex mathematics is based on these fundamental principles.
 

In addition to what we have seen already, there are a number of other logarithmic rules. The following table lists every logarithm rule:

What are The Laws of Logarithms?

 
Mathematical properties known as the laws of logarithms, or logarithmic rules, make logarithmic expressions and equations easier to understand and solve. Since logarithms are the opposite of exponentiation, these laws are predicated on the exponentiation rules.

Product Law

                                                 logb (MN) = logb M + logb N

According to the first law, taking the logarithm after multiplying two numbers together is equivalent to adding their logarithms (of the same base!).

Quotient Law

                                                              logb MN=logb   M-logb   N  

According to the second law, dividing two numbers and taking the logarithm of the result is the same as subtracting their logarithms (again, of the same base).

Power Law 

                                                              logb Mp= plogb M

When a number is raised to a power, the logarithm is equal to the product of the exponent with the logarithm of the base expression. 

Change of Base Law

                                                             loga b = logc blogc a

Because calculators usually only have buttons for logarithms in base 10 (common logarithms, represented by the symbol log) and base e (natural logarithms, represented by the symbol ln⁡), the change of base log law is especially helpful. However, you may come across logarithms in other bases in a variety of mathematical problems.

How to Solve Logarithmic Equations

Although the log rules appear to be relatively easy when viewed in isolation, they are rarely easy to understand during an examination. Usually, you will have to use any number of combinations of logarithmic functions to derive an answer.
Although at first this seems overwhelming, as always, the best technique to handle these issues is to break the question into smaller bits. Solving logarithmic problems uses algebraic manipulation and logarithmic properties. Here is a detailed guide broken out step-by-step:

• Evaluate Logs: To evaluate logs, if the unknown variable is outside the logarithm, you should use the base of the logarithm to find the value of the logarithm itself. For instance, if you have ( log_b(x) = y ), you can express it as ( x = b^y ) to find ( x ). This is pretty simple when the base and the argument of the logarithm are easy to handle, but it might need a calculator for more complex values.

• Convert to Exponential Form: When the unknown variable is located within the logarithm, rewrite the equation as an exponential expression. For instance, if you have the equation (\log_b(x) = y ), you can rewrite it as (x = b^y ). You can find the unknown variable by taking the base of the logarithm and raising it to the power of the other side of the equation.

• Logarithm Combination: When an equation contains multiple logarithms, attempt to combine them by utilizing the properties of logarithms. For instance, you can add or subtract two logarithms with the same base to combine them. As a result, the problem may become simpler. For example, the expression logb(x)  + logb(y) can be written as logb(x) + logb(y) = logb(xy) using the product rule.

• Look for extraneous solutions: Always verify the validity of the solution you have found.  Because they lead to the undefined logarithm of a negative number or zero, some solutions might not be legitimate. These solutions are likely to be discarded as they are referred to as extraneous solutions.

How to Change of Base Rule for Logarithms 

When utilizing calculators that normally only support base 10 (common log) or base 𝑒 (natural log), the Change of Base Rule in logarithms enables you to transform a logarithm with one base into an equivalent expression with a different base. The following is the formula:
 
                                                                   loga b=logc blogc a

In this case, the original base is denoted by 𝑎, the argument by 𝑏, and the new base by 𝑐, which can be any positive number other than 1. For example, you may rewrite log2 8 using base 10 as follows to evaluate it:                                                           
 log 8log 2 0.90310.3010 = 3. Similarly, log3 9, utilizing natural logs, will be In 9In 3  2.19721.0986 = 2. This rule is useful for solving equations, simplifying expressions, and evaluating non-standard bases.
 

Beyond mathematical textbooks we use logarithmic rules in various real-life applications. In this section, we will discuss the application of logarithmic rules 

Intensity of Earthquakes: Richter Scale

In geology, the Richter scale is used to assess earthquake magnitude using logarithmic criteria. Due to the logarithmic nature of this scale, an increase of one unit corresponds to a tenfold increase in the observed amplitude and approximately 31.6 times the energy release. With 𝑀 representing the magnitude, 𝐼 the measured intensity, and 𝐼0 serving as a reference intensity, the formula is M = log10II0. Scientists and emergency personnel can more easily comprehend and convey the magnitude of earthquake activity thanks to this program.

Using the Decibel Scale, sound intensity

Decibels (dB), a unit of measurement for sound intensity, are determined in acoustics using logarithmic methods. This scale appropriately captures how humans perceive loudness because human hearing is logarithmic. The equation that is employed is M = log10II0, where 𝐼0 is the hearing threshold and 𝐼 is the sound's intensity. This scale is useful for audio engineering, noise control, and health safety because a slight increase in decibels results in a noticeable increase in actual sound intensity.

Finance: Growth in Investments and Compound Interest

Logarithmic laws are used in financial mathematics to determine how long it will take for investments to grow through compound interest. Logarithms are used to solve for 𝑡, where A is the final amount, P is the principal, 𝑟 is the interest rate, and 𝑡 is time. The formula that has been rearranged becomes t=log(A/P)log(1+r). 

Population Development and Radioactive Decay

Modeling radioactive decay and population expansion makes extensive use of logarithmic laws. These events have exponential patterns; logarithms allow one to reverse those to find unknown values, such as time. In radioactive decay, for instance, the equation 𝑁(𝑡) = 𝑁0 ⋅ 𝑒⁻ᵏᵗ shows the quantity of a drug declining with time. We employ logarithmic processes to determine the duration of degradation of a material to a specified level. Fields including biology, environmental science, and nuclear physics depend on this tool extensively.