What is the exponential form of the logarithmic equation?

3=log0.60.216
 
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To convert the logarithmic equation \( 3 = \log_{0.6} 0.216 \) into its equivalent exponential form, we use the definition of a logarithm:

\[
\log_b a = c \quad \text{is equivalent to} \quad b^c = a
\]

Given:

\[
3 = \log_{0.6} 0.216
\]

The base \( b = 0.6 \), the exponent \( c = 3 \), and the result \( a = 0.216 \).

So, the exponential form is:

\[
0.6^3 = 0.216
\]

Thus, the answer is \( \boxed{0.6^3 = 0.216} \).

Do you want more details or have any questions? Here are some related questions to deepen your understanding:

1. How can you convert an exponential equation back into logarithmic form?
2. What is the general rule for converting between logarithmic and exponential forms?
3. How would the equation change if the logarithm had a different base, say 2?
4. What does the base of the logarithm tell you about the growth or decay of the exponential function?
5. How can you solve for the base if given a different logarithmic equation?

**Tip:** When converting between logarithmic and exponential forms, remember that the base of the logarithm becomes the base of the exponent in the exponential form.

Learn how to convert a logarithmic equation to exponential form using the definition of logarithms and exponentials. This problem involves logarithms with a base of 0.6 and covers key concepts in logarithms and exponential functions. Suitable for high school students.

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