Solving mathematical expressions involving exponents and logarithms can be intimidating, but with the right approach, it can become manageable. Let's break down the expression "3 to the 4th power 81 in logarithmic form" and solve it step by step.
Understanding the Expression
The expression "3 to the 4th power 81" can be written mathematically as 3^4 = 81. This is an exponential equation, where 3 is the base, 4 is the exponent, and 81 is the result.
Converting to Logarithmic Form
To convert this exponential equation into logarithmic form, we need to use the logarithmic identity:
log_b(a) = c ⇔ b^c = a
where log_b(a) is the logarithm of a to the base b, and c is the exponent.
Using this identity, we can rewrite the exponential equation 3^4 = 81 as:
log_3(81) = 4
Understanding the Logarithmic Identity
The logarithmic identity states that if b^c = a, then log_b(a) = c. This means that the logarithm of a to the base b is equal to the exponent c.
For example, if we have the exponential equation 2^3 = 8, we can rewrite it in logarithmic form as:
log_2(8) = 3
Solving the Logarithmic Equation
Now that we have the logarithmic equation log_3(81) = 4, we can solve for the logarithm.
To solve for the logarithm, we can use a calculator or a logarithmic table. However, in this case, we can also use the fact that 81 is a perfect cube (3^4 = 81).
Using this fact, we can rewrite the logarithmic equation as:
log_3(3^4) = 4
Using the logarithmic identity, we can simplify this equation to:
4 = 4
This shows that the logarithmic equation is true, and we have solved it.
Practical Applications of Logarithms
Logarithms have many practical applications in science, engineering, and finance. They are used to model population growth, chemical reactions, and financial transactions.
For example, in finance, logarithms are used to calculate compound interest and investment returns. They are also used in science to model the growth of populations and the spread of diseases.
Common Logarithmic Identities
There are several common logarithmic identities that are used to simplify and solve logarithmic equations. These include:
- log_b(b) = 1
- log_b(1) = 0
- log_b(a) + log_b(c) = log_b(ac)
- log_b(a) - log_b(c) = log_b(a/c)
These identities can be used to simplify and solve logarithmic equations, and are essential for working with logarithms.
Conclusion
In conclusion, solving the expression "3 to the 4th power 81 in logarithmic form" involves using the logarithmic identity to convert the exponential equation into logarithmic form, and then solving for the logarithm. Logarithms have many practical applications, and understanding the common logarithmic identities is essential for working with logarithms.
We hope this article has helped you understand how to solve the expression "3 to the 4th power 81 in logarithmic form" and has provided you with a deeper understanding of logarithms and their applications.
Share Your Thoughts
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Related Articles
- How to Solve Exponential Equations
- Understanding Logarithmic Identities
- Practical Applications of Logarithms
FAQs
What is the logarithmic identity?
+The logarithmic identity states that if b^c = a, then log_b(a) = c.
How do I solve a logarithmic equation?
+To solve a logarithmic equation, you can use a calculator or a logarithmic table, or you can use the logarithmic identity to simplify and solve the equation.
What are some common logarithmic identities?
+Some common logarithmic identities include log_b(b) = 1, log_b(1) = 0, log_b(a) + log_b(c) = log_b(ac), and log_b(a) - log_b(c) = log_b(a/c).