Question
\( 1 \leftarrow \) Change the exponential statement to an equivalent statement involving a logarithm. The equivalent logarithmic statement is \( \square \) (Type an equation.)
Ask by Wade Mitchell. in the United States
Feb 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The equivalent logarithmic statement is \( \log_a(1) = 0 \).
Solution
To change the exponential statement \( 1 \) to an equivalent logarithmic statement, we need to understand the relationship between exponents and logarithms.
The general form of an exponential statement is:
\[
a^b = c
\]
This can be rewritten in logarithmic form as:
\[
\log_a(c) = b
\]
In your case, since the statement is simply \( 1 \), we can express it in terms of logarithms. However, we need a base and an exponent to create a meaningful logarithmic statement.
Assuming we want to express \( 1 \) as an exponent of any base \( a \) (where \( a > 0 \) and \( a \neq 1 \)), we can say:
\[
a^0 = 1
\]
This can be rewritten in logarithmic form as:
\[
\log_a(1) = 0
\]
Thus, the equivalent logarithmic statement is:
\[
\log_a(1) = 0
\]
You can choose any base \( a \) for this logarithmic statement. If you have a specific base in mind, please let me know!
Answered by UpStudy AI and reviewed by a Professional Tutor
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The Deep Dive
To transform the exponential statement \( a^b = c \) into logarithmic form, it becomes \( \log_a(c) = b \). So, if your exponential statement is, say, \( 2^3 = 8 \), the equivalent logarithmic statement would be \( \log_2(8) = 3 \). This relationship is super handy! Logarithms help us unravel exponential growth—think of them as the 'undo' button for powers. You can use them in anything from compound interest calculations to analyzing the decay of radioactive materials—where knowing the exponent can lead to some key insights.