Rewrite the equation in exponential form. loga(b) = c
The Correct Answer and Explanation is:
The exponential form of the logarithmic equation
logₐ(b) = c
is:
aᶜ = b
Explanation
Understanding how to convert between logarithmic and exponential forms is essential in algebra and higher-level mathematics. The equation logₐ(b) = c is a logarithmic expression. It asks the question: To what power must the base “a” be raised to get the number “b”? The answer to that question is “c”.
To rewrite this in exponential form, we must translate the meaning directly into an equation that uses exponents. In this case, since “a” raised to the power “c” equals “b”, the corresponding exponential form becomes:
aᶜ = b
This conversion is not just a mathematical rule but reflects the core relationship between exponents and logarithms. Think of logarithms as the inverse operations of exponentiation. Just as subtraction undoes addition and division undoes multiplication, logarithms undo exponentiation.
Let us consider a concrete example to make the concept clearer:
Suppose you have the equation:
log₂(8) = 3
This equation is asking: To what power must 2 be raised to give 8? The answer is 3, because 2³ = 8. Therefore, the exponential form is:
2³ = 8
The logarithmic and exponential forms represent the same relationship, just written differently. Switching between these forms allows you to solve equations depending on what is known and what is unknown. For example, if the exponent is unknown, logarithmic form may help you find it. If the exponent is given, exponential form allows you to calculate the value of the expression.
Understanding this concept provides a foundation for solving logarithmic and exponential equations in algebra, calculus, and many real-world applications such as compound interest, population growth, and radioactive decay.
