Mastering Logarithms: Change of Base Formula and Solving Techniques
Comprehensive Definition, Description, Examples & Rules
Introduction
The most important tools in mathematics for solving exponential equations and computing the relationship between numbers are logarithms. Logarithms are simply the opposite of exponentiation and help us to figure out the exponent to which a particular base number must be raised or multiplied by itself given several times to achieve a particular result. In other words, logarithms help us ease up difficult mathematical expressions containing exponents and make our problem-solving procedure more ordered and organized.
The change of base formula comes in handy when we have to compute values for logarithms in different bases. Using the formula, we can convert logarithms from one base to another. This attribute of logarithms is highly useful in various applications we come across in our everyday lives like engineering, physics, finance, and many more.
Logarithms and Their Basics
Logarithms are those mathematical tools or functions that help us discern how exponents relate and work with each other. They help us identify the number of times a base number must be multiplied by itself to attain a defined result, or in other words, they help us find out the value of the exponent or power to which a base number is raised.
Some of the properties of logarithms are:
- Product Rule: logₐ mn = logₐ m + logₐ n
- Quotient Rule: logₐ m/n = logₐ m – logₐ n
- Power Rule: logₐ mn = n logₐ m
- Change of Base Rule: logb a = (log꜀ a) / (log꜀ b)
Logarithms are employed in a variety of fields like:
- Engineering: Engineers use logarithms to design electrical circuits, process signals, and more.
- Economics: Economists use logarithms to compute compound interests, analyze investment opportunities, etc.
- Biology: biologists employ the log base change formula in various processes like gene mapping, modeling, etc.
Among the many logarithm bases, here are some common logarithm bases:
- Base 10: Base 10 is represented as log10 (x) or log(x). It is highly compatible with the decimal system which is why it is used in numerous everyday applications.
- Natural Logarithm: Natural logarithm is represented as ln(x). It is based on the constant ‘e’ which is an irrational number valuating up to 2.71828. It is used extensively in calculus.
Change of Base Formula
The change of base rule or formula is a crucial concept in mathematics as it helps one to change the base of logarithms from one to another, as is suggested by its name. The formula for positive numbers a, b, and c, is written as:
logb a = logc (a) / logc (b)
In this formula,
- logb (a): Logarithm of a to the base b
- logc (a): Logarithm of a to the base c
- logc (b): Logarithm of b to the base c
The change of base formula is useful and important when dealing with complex calculations and problems that involve bases that need to be changed from one to another to compute adequate results. The formula helps us to devise logarithms about a variety of bases.
How to Solve for Logarithms with Different Bases
Here is a step-by-step instructions on solving logarithmic equations when the base is other than 10 or e.
- Step 1: Transpose all the terms that don’t involve logarithms to one side of the equation.
- Step 2: Change the logarithm to its exponential form.
- Step 3: Convert the equation to its simplest form.
- Step 4: Use common techniques of algebra to single out variables.
Have a look at the following examples with varying bases to understand the procedure better:
- Solve log4(x) = 2
Solution: employing the inverse operation, we get 4^2 = x
Hence, x = 16
- Solve log6 (x) = 1 + log6 (2x)
Solution: By transposing the entities on one side we get, log6 (x) – log6 (2x) = 1
log6 (x/2x) = 1
log6 (½) = 1
Employing the inverse operation, x = ½.
Some real-world applications where solving logarithmic equations is essential are:
- Logarithm equations are used to compute compound interest in finance.
- Logarithm equations help us analyze the growth and spread of diseases and outbreaks.
- Logarithm equations also help us discern, analyze, and make accurate and precise predictions about various population trends.
Converting Logarithmic Bases
The change of base formula helps in converting log bases from one to another. It comes in handy in situations where we do not know the base of a certain logarithm by providing us with a specified rule and procedure to follow. The formula for positive numbers a, b, and c, is written as:
logb a = logc (a) / logc (b)
In this formula,
- logb (a): Logarithm of a to the base b
- logc (a): Logarithm of a to the base c
- logc (b): Logarithm of b to the base c
Here is a practical example of converting logarithms from one base to another:
- Solve: log64 8
Solution: log64 8 = [log 8] / [log 64]
= [log 8] / [log 8^2]
= [log 8] / [2 log 8] [∵ log a^m = m log a]
= 1 / 2
The benefits of converting logarithmic bases are:
- The formula gives us a certain standard set of rules to be applied when computing varied logarithmic bases.
- It helps us ease up complex calculations which include varied bases.
- It helps us employ logarithmic functions in a varied range of contexts.
Change of Base Exponents
The opposite of the change of base formula in logarithms is the function of change of base exponents. It helps us represent an exponential function with a different base which comes in handy in the process of easing up the calculations.
The formula and technique used to change the base of exponential expressions is given as:
a^x=[b^logb(a)]^x=b^x⋅logb(a)
Hence, any exponential function with a as a base can be written as an exponential function with b as a base by incorporating the exponent with the logarithm of the new base accordingly.
Here are some examples from the real world that use change of base in exponential expressions:
- Population growth and decay can be discerned using the formula.
- The decay of radioactive elements in physics can also be analyzed through the formula.
How to Solve Logarithms with Base 10 and Base e
Here are step-by-step instructions on solving logarithmic equations with base 10 and base e (natural logarithm).
- Logarithmic equations with Base 10
- Transpose all the terms that don’t have a logarithm to one side of the equation.
- Turn the logarithms to their exponential forms.
- Simplify the equation.
- Use algebra to single out variables.
How to solve log base 10: Solve log7(x) = 2
Solution: employing the inverse operation, we get 7^2 = x
Hence, x = 49
- Logarithmic equations with Base e
- Transpose all the terms that don’t have a logarithm to one side of the equation.
- Turn the logarithms to their exponential forms.
- Simplify the equation.
- Use algebra to single out variables.
Example: what is x in ln(x) = 3
Solution: ln(x) = 3
e^3 = x
x = 20.0855 (approximately)
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Key Takeaways
- Logarithms are simply the opposite of exponentiation and help us to figure out the exponent to which a particular base number must be raised or multiplied by itself given several times to achieve a particular result.
- The change of base formula for positive numbers a, b, and c, is written as logb a = logc (a) / logc (b)
- Logarithmic expressions and formulas find use in everyday applications like physics, engineering, biology, economics, and many more.
Quiz
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Frequently Asked Questions
You can use the change of base formula when you want to change the base of the logarithm from one base to another to ease up complex calculations as the change of base calculator only mentions logarithm functions of common logarithmic bases like log base 10.
The Common Logarithm (Base 10) and the Natural Logarithm (Base e) are two commonly used logarithms in mathematics used in various applications in the real world.
To solve logarithmic equations with different bases, you can use the change of base formula.
To solve logarithmic equations, follow the given steps:
- Transpose all the terms that don’t have a logarithm to one side of the equation.
- Turn the logarithms to their exponential forms.
- Simplify the equation.
- Use algebra to single out variables.
The significance of the change of base exponents is that it is the opposite function of the change of base of logarithms. It helps us represent an exponential function with a different base which comes in handy in the process of easing up the calculations.
